Welcome to the 6th COLOGNE TWENTE WORKSHOP ON GRAPHS AND COMBINATORIAL OPTIMIZATION University of Twente, Enschede, The Netherlands 29-31 May, 2007 This volume collects the extended abstracts of the contributions that have been selected for presentation at the Workshop. As it was the case with previous workshops, we are planning to edit a special, fully refereed, proceedings volume with Discrete Applied Mathematics or Discrete Optimization and hereby invite all participants to submit full-length papers relating to the topics of the Workshop. Deadline of submission is: 15 September, 2007. Scientific Committee U. Faigle (Cologne), J.L. Hurink (Enschede), F. Maffioli (Milan), R. Schrader (Cologne), R. Schultz (Duisburg). Organizing Committee J.L. Hurink (Chair), W. Kern, G.F. Post, G.J. Still, H.E. Baarsma, D.R. Grigoras, J.J. Paulus, X. Wang. We thank Dini Heres-Ticheler for her help. We gratefully acknowledge the financial support from the NWO (the Organisation of Scientific Research in the Netherlands), CTIT (Centre for Telematics and Information Technology), Stichting Universiteits Fonds and the University of Twente. J.L. Hurink W. Kern G.F. Post G.J. Still Contents Extended abstracts N.M.M. de Abreu, P. Hansen, C.S. Oliveira, L.S. de Lima Bounds on the index of the Signless Laplacian of a graph involving the average degree of neighbors of a vertex 1 N.E. Aguilera, V.A. Leoni, G.L. Nasini Some flexibility problems and their complexity 5 S.D. Andres Directed defective asymmetric graph coloring games on forests 9 H.L. Bodlaender, A. Grigoriev, N.V. Grigorieva, A. Hendriks The Valve Location Problem 13 P.S. Bonsma Most balanced minimum cuts and partially ordered knapsack 17 A.B.A. Ceselli, G. Righini 23 A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint S. Coene, F.C.R. Spieksma A latency problem with profits 29 U. Faigle, B. Peis A two-phase greedy algorithm for modular lattice polyhedra 33 J.F. Feng 39 A characterization for jump graphs containing complementary cycles H. Fernau Dynamic Programming for Queen Domination 43 R. Gollmer, U. Gotzes, F. Neise, R. Schultz 49 Stochastic Programs with Dominance Constraints Induced by Mixed-Integer Linear Recourse M.C. Golumbic, M. Lipshteyn, M. Stern Edge intersection graphs of single bend paths on a grid 53 A. Grigoriev, J. van Loon, M. Sviridenko, M. Uetz, T. Vredeveld Optimal Bundle Pricing for Homogeneous Items 57 E.W. Hans Operations Research for hospital process optimization 61 I.B.A. Hartman On Path Partitions and Colorings in Digraphs 63 C. Heilporn, M. Labbé, P. Marcotte, G. Savard On a Network Pricing Problem with Consecutive Toll Arcs 67 B. Heydenreich, R. Müller, M. Uetz, R. Vohra On Revenue Equivalence in Truthful Mechanisms 69 S. Hossain, M.F. Zibran 73 A Multi-phase Approach to the University Course Timetabling Problem J.L. Hurink, T. Nieberg Approximating Minimum Independent Dominating Sets in Wireless Networks 77 Y. Kempner, V.E. Levit Representation of Poly-antimatroids 81 W. Kern, X. Wang On full components for Rectilinear Steiner tree 85 J. Leblet, J.X. Rampon St-serie decomposition of orders 89 R. Li, S. Li, J. Feng The number of vertices whose out-arcs are pancyclic in 2-strong tournaments 95 S. Li, W. Meng, Y. Guo A local tournament contains a vertex whose out-arcs are g-pancyclic 99 L. Liberti 103 A useful characterization of the feasible region of binary linear programs V. Lozin, M. Milanic̆ 107 On the maximum independent set problem in subclasses of planar and more general graphs D. Lozovanu, S. Pickl 111 Multiobjective Hierarchical Control of Time-Discrete Systems and Determining Stackelberg Strategies G. Nannicini, Ph. Baptiste, D. Krob, L. Liberti 115 Fast point-to-point shortest path queries on dynamic road networks with interval data J.J. Paulus, J.L. Hurink Decomposition Method for Project Scheduling with Adjacent Resources 119 A. Peeters, K. Coolsaet, G. Brinkmann, N. Van Cleemput, V. Fack 123 GrInvIn for Graph Theory Teaching and Research M.C. Plateau, L. Liberti, L. Alfandari Edge cover by bipartite subgraphs 127 p J.A. Rodrı́guez-Velázquez, J.M. Sigarreta On the defensive k-alliance number of a graph 133 J. Spoerhase, H.-C. Wirth 137 Relaxed Voting and Competitive Location on Trees under Monotonuous Gain Functions Bounds on the index of the Signless Laplacian of a graph involving the average degree of neighbors of a vertex Nair Maria Maia de Abreu a a Federal University of Rio de Janeiro, Brazil Pierre Hansen b b GERAD, HEC Montreál Carla Silva Oliveira c c School of Statistic Sciences, Brazil Leonardo Silva de Lima d d Federal Center of Technological Education, RJ, Brazil Key words: average degree, index, signless Laplacian, bounds 1 Introduction Let G = (V, E) be a simple graph with n vertices vi ∈ V and m edges {vi , vj } ∈ E, for i, j = 1, 2, . . . , n and i = j. In this case, vi is adjacent to vj and we denote vi ∼ vj . The vertex degree is dvi , and the degree sequence of G is d(G) = (dv1 , dv2 , · · · , dvn ), where dv1 ≥ dv2 ≥ · · · ≥ dvn . The maximum degree of G is ∆(G) = dv1 , the minimum degree is δ(G) = dvn , the average  degree of G is d = 1,...,n dni and the average degree of the neighbors of vi is  mi = d1i vj ∼vi dj , [8]. The eigenvalues of G are the eigenvalues of the adjacency matrix A(G), given as λ1 ≥ . . . ≥ λn−1 ≥ λn , where λ1 is called the index of G. The Laplacian matrix of G is L(G) = D(G) - A(G), where D(G) Email addresses: nair@pep.ufrj.br (Nair Maria Maia de Abreu), Pierre.Hansen@gerad.ca (Pierre Hansen), carlasilva@ibge.gov.br (Carla Silva Oliveira), llima@cefet-rj.br (Leonardo Silva de Lima). 1 is the diagonal matrix of vertex degrees of G. Its eigenvalues are displayed as µ1 ≥ . . . ≥ µn−1 ≥ µn and µ1 is the index of Laplacian matrix. Since L(G) and A(G) are well known, there are many results to their spectrum, see Cvetković and Rowlinson [3] and, Merris [11]. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian of G and rarely appears in the literature. The eigenvalues of signless Laplacian are denoted as q1 ≥ q2 ≥ · · · ≥ qn . Recently, Cvetković et al. [4] suggested the investigation of properties related to the spectrum of Q(G) since they conjecture that when the number of vertices of a graph G is sufficiently large, it is possible, with a high probability, to characterize G by the spectrum of Q(G). For every G, it is well known that L(G) is a positive semidefinite and singular matrix. So, µn = 0 and, if G is a non connected graph, µn−1 = 0, Merris [11]. Although Q(G) is a positive semidefinite matrix, it is not necessary singular. When it happens, G is a bipartite graph. Further, if G is connected then qn = 0 is a simple eigenvalue [4]. Many, but not all, of the lower and upper bounds on the index of Laplacian are shown to be valid also for the index of the signless Laplacian. Moreover, such bounds may also be obtained by doubling some bounds on the index of the adjacency matrix. In this paper, we determine new bounds to the index of Q(G) which are relative to the average degree of neighbors of a vertex and the average degree of a graph G. A few results about eigenvalues of the signless Laplacian are available in the literature. Cvetković et al. [4] discussed the importance to investigate the spectrum of Q(G) and proved some results as follows: k-regular graphs can be recognized from the spectrum of Q(G) and, in this case, q1 = 2k; the spectra of the Laplacian and signless Laplacian of bipartite graphs define the same set; when vi ∼ vj , the min(di + dj ) and the max(di + dj ) are a lower and an upper bounds to q1 , respectively; thresholds graphs among others have the maximum value to q1 . Yan [12] determined the spectrum of Q(G) on certain families of graphs as complete graphs, complete bipartite graphs and complete graphs without an edge. In the same paper, he determined a sharp lower and upper bounds to q1 on simple connected graphs and improved them for trees. Using AGX system, Mustapha and Hansen [1] proposed some conjectures involving q1 , q2 , δ(G), ∆(G), the average degree and others invariants. Some of these conjectures are proved in Cvetković et al. [5] while the others remain open. 2 New bounds to the index of signless Laplacian Based on the similar techniques developed by Brankov et al. [2] who determined bounds on the larger eigenvalue of the Laplacian matrix, we obtained two upper bounds to the index of the signless Laplacian involving degrees and the average degrees of neighbors of a vertex on G. 2 Theorem 1 Let G be a simple and connected graph. Then  q1 (G) ≤ maxvi 2dvi (dvi + mvi ). and q1 (G) ≤ maxvi dvi + (1)  (dvi )2 + 8(dvi mvi ) . 2 (2) According to the techniques used by Li and Zhang [10] and Zhang [13] to prove bounds to the index of Laplacian of graph, we obtained the following results. Theorem 2 Let G be a graph. Then, q1 (G) ≤ maxvi ∼vj { dvi (dvi + mvi ) + dvj (dvj + mvj ) }. dvi + dvj (3) Theorem 3 Let G be a simple and connected graph. Then q1 (G) ≤ max{dvi +  dvi mi | vi ∈ V (G)}. (4) The equality holds, if G is a k−regular, a bipartite regular or a semi-regular graph. Further, using the known matrix result min1≤i≤n n  qij ≤ q1 (G) ≤ max1≤i≤n j=1 n  qij j=1 n n q xx i=1 j=1 ij i j , n 2 and the Rayleigh’s quotient, R = x i=1 i bound that is given in the following theorem. we obtained another new Theorem 4 Let G be a graph with n vertices. Then, 2δ ≤ 2d¯ ≤ q1 ≤ 2∆. Finally, to get the proofs of the last two theorems, we have to apply Theorems 2.3 and 2.4 from Zhang [13] and use a basic relation from Cvetković et al. [4]. This relation involves the characteristic polynomials of the signless Laplacian matrix of a graph and the adjacency matrix of its line graph. Theorem 5 Let G be a simple and connected graph and LG be the line graph of G. Then  q1 (G) ≤ maxvi ∼vj { dvi (dvi + mvi ) + dvj (dvj + mvj )}, 3 with the true equality if and only if G is either a regular or a semi-regular graph. Moreover, q1 (G) ≤ maxvi ∼vj {2 +  dvi (dvi + mvi − 4) + dvj (dvj + mvj − 4) + 4}. In this case, the equality holds if and only if G is either P4 , a regular or a semi-regular graph. References [1] M. Aouchiche and P. Hansen, Some conjectures about signless Laplacian matrices, 2006. [2] V. Brankov, P. Hansen, D. Stevanović, Automated conjectures on upper bounds for the largest Laplacian eigenvalue of graphs, Linear Algebra and its Applications, 2006, (414), 407-424. [3] D. Cvetković and P. Rowlinson, The largest eigenvalue of a graph: a survey, Linear and Multilinear Algebra, 1990, (28), 3-33. [4] D. Cvetković, P. Rowlinson and S. Simić, Signless Laplacian of finite graphs, 2007,doi:10.1016/j.laa.2007.01.009. [5] D. Cvetković, P. Rowlinson and S. Simić, Bounds for the signless Laplacian eigenvalues, to be presented on The First IPM Conference on Algebraic Graph Theory, Tehran, Iran, 21-16 April, 2007. [6] K.C. Das, A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs, Linear Algebra and its Applications, 2004, (376), 173-186. [7] J.M. Guo, A new upper bound for the Laplacian spectral radius of graphs, Linear Algebra and its Applications, 2005, (400), 61-66. [8] R. Diestel, Graph Theory, GTM 173, Springer, 1997. [9] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [10] J.S. Li, X.D. Zhang, On the Laplacian eigenvalues of a graph, Linear Algebra and its Applications, 1988, (285), 305-307. [11] R, Merris, Laplacian matrices of graphs: a survey, Linear Algebra and its Applications, 1994, (197/198), 143-176. [12] C. Yan, Properties of spectra of graphs and line graphs, Appli. Math. J. Chinese Univ. Ser.B, 2002, (3), 371-376. [13] X.D. Zhang, Two sharp upper bounds for Laplacian eigenvalues, Linear Algebra and its Applications, 2004, (376), 207-213. 4 Some flexibility problems and their complexity Néstor E. Aguilera a a CONICET and UNL. IMAL/INTEC, Güemes 3450, 3000 Santa Fe, Argentina Valeria A. Leoni, Graciela L. Nasini b b CONICET and UNR, Av. Pellegrini 250, 2000 Rosario, Argentina Key words: exchanger networks, flexibility problems, computational complexity 1 Preliminaries The concept of flexibility arose from chemical engineering problems in the design of heat exchanger networks (see [1] and [3]). An exchanger network is a network D = (V1 ∪ V2 , A) with V1 ∩ V2 = ∅, A ⊆ V1 × V2 and quantities given by c ∈ RV1 ∪V2 . If i ∈ V1 (i ∈ V2 ) , ci is the supply (demand ) of vertex i. A flow in D is a function f : A → R+ verifying P j:(i,j)∈A fij ≤ ci ∀i ∈ V1 and P i:(i,j)∈A fij ≤ cj ∀j ∈ V2 . The value of f is defined as val(f ) = (i,j)∈A fij , and f is a maximum flow if there is no flow f ′ in D such that val(f ′ ) > val(f ). P Clearly, the problem of finding the maximum flow in D can be formulated as a maximum flow problem over an extended network G = (V, A′ ) with a source s and a sink r, where V = {s} ∪ V1 ∪ V2 ∪ {t}, A′ = ({s} × V1 ) ∪ A ∪ (V2 × {t}) and the capacity function c′ over A′ defined as Email addresses: aguilera@ceride.gov.ar (Néstor E. Aguilera), valeoni@fceia.unr.edu.ar, nasini@fceia.unr.edu.ar (Valeria A. Leoni, Graciela L. Nasini). 5 c′si = ci for i ∈ V1 , c′ij = ∞ for (i, j) ∈ A and c′jt = cj for j ∈ V2 . In this context, we may apply the classical max flow-min cut results due to Ford and Fulkerson [4]. Given a network G = (V, A) with source s and sink r, the capacity function ∗ ∗ given by c ∈ RA + and A ⊆ A, consider the subnetwork G induced by the ∗ ∗ elements in A . We denote by ξ(c) and ξ (c), the maximum flow value over G and G∗ , respectively. The Maximum Flow Flexibility Problem (FF ) is formally formulated as ∗ INSTANCE: G = (V, A); s, r ∈ V ; c+ , c− ∈ ZA +; A ⊆ A − + ∗ QUESTION: Is there c ∈ RA + with c ≤ c ≤ c and ξ (c) < ξ(c) ? If we consider an instance of the FF problem where A∗ = {e} ⊆ A, c− = 0 and c+ = ∞, the question in the FF problem is equivalent to asking whether the edge e is useful [2], i.e. whether removing e, the value of the maximum flow changes. The NP -completeness of the FF problem and the problem of deciding if a given edge is useful have been studied in [8] and [2], respectively. In both cases, the proof is based on the reduction of the Two Direct Vertex Disjoint Path (DVDP2 ) problem. 2 The complexity of the FT problem We now consider the FF problem restricted to instances over exchanger networks. We will call this problem the FT problem. In contrast to the mentioned result on the FF problem, in this case we have the following Lemma 2.1 If c− = 0, the FT problem can be solved in polynomial time. However, for the general case we proved the following Theorem 2.2 The FT problem is strongly NP-complete. The proof of the above result is based on the reduction of the Balanced Complete Bipartite Graph (BCBG) problem to the FT problem. The BCBG problem consists of finding a complete bipartite balanced subgraph of certain size in a given bipartite graph. This problem is strongly NP -complete [6, p. 196]. The key in our study of the computational complexity of the FT problem was the identification of what we call test sets. A test set is a finite subset of states 6 − + W ⊆ S = {c ∈ RA + : c ≤ c ≤ c } such that flexibility in W implies flexibility in S. We omit in this abstract the results obtained on test sets. 3 Flexibility and combinatorial optimization problems We have seen that, in spite of the fact that the Maximum Flow Problem in a network is a polynomial problem, the associated flexibility problem is N P complete. We are interested in polynomial flexibility problems. In this section we extend the flexibility concept to a general combinatorial optimization problem. Let A denote a finite set, c ∈ RA subsets + a vector of costs, and F(A) a family ofP of A. The cost of an element F ∈ F(A) will be indicated by c (F ) = a∈F ca . Moreover, ξ(c) = min(max) {c(F ) : F ∈ F(A)}. If F(A) = ∅, we will consider ξ(c) = +∞ and ξ(c) = −∞, respectively. We will identify a particular combinatorial optimization problem by the oracle algorithm F which decides in constant time, whether a given subset of A belongs to the family F(A) [7, p. 26]. Let us consider a family of instances of F defined by a given set A and costs o n in S = c ∈ RA : c− ≤ c ≤ c+ , with c− , c+ ∈ ZA +. For A∗ ⊆ A, consider the set of instances of F defined by A∗ and the “pro∗ jection” of c in S into RA . We will denote the optimum value of the corresponding optimization problem over F(A∗ ) by ξ ∗ (c). Given a combinatorial optimization problem F, we define the F flexibility problem as follows: INSTANCE: (A, F) ; A∗ ⊆ A ; c+ , c− ∈ ZA + − + ∗ QUESTION: Is there c ∈ RA + with c ≤ c ≤ c and ξ(c) 6= ξ (c) ? We restrict ourselves to the family of combinatorial problems F for which the ∗ optimal value is monotone under inclusion, i.e., for every c ∈ ZA + , ξ(c) ≤ ξ (c) if F is a minimization problem, and ξ(c) ≥ ξ ∗ (c) if F is a maximization problem. In this wide family of combinatorial problems we first prove that, given a combinatorial optimization problem, the associated flexibility problem is always at least as hard as the optimization problem in the following sense. 7 Theorem 3.1 If the F flexibility problem is solved in polynomial time, then so is F. The results on the FT problem shown in the previous section suggest that the converse in theorem 3.1 is not true. The following result confirms this idea. Theorem 3.2 The Shortest Path flexibility problem is NP-complete. The proof is also based on the reduction of the Two Direct Vertex Disjoint Path (DVDP2 ) problem. Thus, when looking for polynomial time flexibility problems, we should start with an easy base problem. For example, through their characterization by greedy algorithms, matroids form a large family of such problems, as the following theorem shows: Theorem 3.3 The maximum independent set (in a matroid) flexibility problem is a polynomial problem. Once again, results on test sets were fundamental for the study of the computational complexity of F flexibility problems. We also omit them in this abstract. References [1] N. Aguilera and G. Nasini, Flexibility Test for Heat Exchanger Networks with Uncertain Flowrates. Comp. Chem. Engng., Vol. 19, No. 9, pp. 1007–1017, 1995. [2] T. C. Biedl, B. Brejová and T. Viñar, Simplifying Flow Networks. M. Nielsen and B. Rovan (Eds.): MFCS 2000, LNCS 1893, pp. 192–201, 2000. [3] J. Cerdá, M. R. Galli, N. Camussi and M. A. Isla, Synthesis of Flexible Heat Exchanger Networks-I. Convex Networks. Comp. Chem. Engng., Vol. 14, No. 2, pp. 197–211, 1990. [4] L. R. Ford and D. R. Fulkerson, Flows in Networks. Princeton University Press, 1962. [5] S. Fortune, J. E. Hopcroft and J. C. Wyllie, The Directed Subgraph Homeomorphism Problem. Theor. Comp. Sci., Vol. 10, pp. 111–121, 1980. [6] M. R. Garey and D. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., 1979. [7] M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988. [8] G. L. Nasini, Sobre la Flexibiliad en Problemas de Optimización. Tesis Doctoral, Universidad Nacional de Rosario, Argentina, 1997. 8 Directed defective asymmetric graph coloring games on forests Stephan Dominique Andres Zentrum für angewandte Informatik Köln, Universität zu Köln, Weyertal 80, 50931 Köln, Germany Key words: game chromatic number, dichromatic number, graph coloring game, digraph, directed forest, defective coloring, asymmetric competitive coloring 1 Introduction We consider the following 2-person game which is played by the players, Alice and Bob, on a digraph D that is uncolored at the beginning, with a color set C, and nonnegative integers d, a, and b. The players alternately color uncolored vertices of D with colors from C until this is not possible any more by one of the following rules. While a color can be used for several vertices, a vertex can have only one color. Alice colors a vertices in a turn, Bob b vertices. However, if at the beginning of the last move of Alice (resp. Bob) only x ≤ a (resp. x ≤ b) uncolored vertices are left, Alice (resp. Bob) has to color only x vertices. Whenever a player colors a vertex v with color i, then every in-arc (w, v) is deleted in D except for those in-arcs (w, v) for which w has been colored with i before. For any color i, the remaining arcs in the subdigraph induced by the vertices of color i form the defect digraph Di of color i. The main rule the players have to respect is that at any state of the game for any color i the defect digraph Di has maximum total degree at most d. (If they cannot respect this rule they cannot move any more.) Here, the total degree of a vertex v is the sum of the in-degree of v and the out-degree of v (not counting loops.) The maximum total degree is the maximum of the total degrees of all vertices. Alice wins if every vertex is colored at the end of the game (or if a = b = 0), otherwise Bob wins. In order to make the game well-defined we Email address: andres@zpr.uni-koeln.de (Stephan Dominique Andres). URL: http://www.zaik.uni-koeln.de/∼andres (Stephan Dominique Andres). 9 will assume that Alice has the first move, and missing a turn is not allowed. This game is called (directed) (a, b)-coloring game with defect d or d-relaxed (directed) (a, b)-coloring game. The smallest cardinality n = #C of a color set C for which Alice has a winning strategy for the directed (a, b)-coloring game with defect d played on the digraph D is called d-relaxed (a, b)-game chromatic number (a,b) χdg (D). If graphs are considered as digraphs with pairs of opposite arcs instead of edges, then the parameter described above generalizes a lot of parameters of competitive graph coloring. For example, for a graph G, (1,1) χ0g (G) is the game chromatic number introduced by Bodlaender [3]. In the same way, (a,b) χ0g (G) is the (a, b)-game chromatic number introduced by Kierstead [5]. (1,1) χdg (G) is the d-relaxed game chromatic number introduced by Chou et al. [4], and (a,0) 0 χg (G) is the usual chromatic number for a ≥ 1. But this parameter also generalizes some digraph concepts. For a digraph D, is the game chromatic number of a digraph introduced by the author [1], and (a,0) χ0g (D) is the dichromatic number of D for a ≥ 1 which was introduced by Neumann-Lara [6]. (1,1) 0 χg (D) For a nonempty class C of digraphs we further define (a,b) d χg (C) 2 = sup (a,b) χdg (D). D∈C Main results  be the class of orientations of forests and F be the class of undirected Let F forests, considered as digraphs as above. Theorem 1 (a,b) d  χg (F) =  b d+1  In particular, Theorem 1 means Theorem 2  b d+1  + 2 if a ≥ b ≥ 1. (1,1) d  χg (F ) + 2 ≤ (a,b) χdg (F ) ≤  = 2 for d ≥ 1. b d+1  + 3 for a ≥ b ≥ 1. The lower bounds for both Theorems are obtained easily. We will give a sketch of proof for the upper bound of Theorem 1. Let F be an orientation of a forest and c = ⌊b/(d + 1)⌋ + 2. We prove that Alice has a winning strategy for the d-relaxed (a, b)-coloring game on F with c colors. During the game, by the arc deletion rule of the game, the forest is subdivided into more and more connected components which we call trunks. 10 Alice’s winning stategy consists in guaranteeing that at the end of her move every trunk has only vertices of one color, and these colored vertices of a trunk are connected (without uncolored vertices inbetween). After a move of Bob, in several trunks there may be additional colored vertices (up to b in total). Alice writes down the vertices colored by Bob in a list v1 , v2 , . . . , vb . Then, when she colors the k-th vertex of her next move, she pretends that Bob has colored only the vertices v1 , v2 , . . . , vk before. So, inductively we may assume that in her model there is at most one trunk which does not obey her invariant, in particular this is the trunk with vk and possibly a connected set C0 of colored vertices of one color. If there is no trunk conflicting with her invariant, Alice may simply color an uncolored sink in a trunk. It is easy to see that, by the rules of the game, an uncolored sink exists unless all vertices are colored. On the other hand, if there is a trunk with C0 and vk not obeying Alice’s invariant, we consider the shortest path P from C0 to vk . Since the first arc in P is directed away from C0 and the last arc in P is directed away from vk , there is an uncolored sink x in P (which is not necessarily a sink in F ). Alice colors x except if this is not possible in case x is one of Bob’s vertices vk+1 , . . . , vb . In this exceptional case, or if she has to color additional vertices, Alice again colors a global sink. If she plays this way, the trunk is broken in at least two parts, both obeying her invariant. Alice can react on every vertex vk of Bob since a ≥ b. By induction, at the end of her move, there will be no unregarded vertices of Bob any more, and Alice has reinstalled her invariant. Moreover we have to prove that Alice always finds a feasible color in order to make her move. Therefore we define the color weight g(v) of a colored vertex v as the cardinality of the neighbors of v colored with the same color as v plus one. The weight G(w) of an uncolored vertex is G(w) =  g(v), v∈Ncol (w) where Ncol (w) are the colored neighbors of w. It is obvious (by the pigeonhole-principle) that a vertex w can be colored feasibly if G(w) ≤ (d + 1)c − 1. (1) Let (d + 1)i ≤ b ≤ (d + 1)(i + 1) − 1. Then c = i + 2. After a move of Bob, an uncolored vertex w can be neigbored to a colored vertex that has been there before Bob’s move and has therefore color weight ≤ (d + 1), and the weight of w can be increased by b because of the b vertices Bob has colored. Therefore after Bob’s move G(w) ≤ d + 1 + b ≤ (d + 1)c − 1, 11 i.e. (1) holds. To finish the proof one has to show that (1) also holds during a move of Alice. This can be done with a careful contradiction argument in the case d ≥ 1. (One has to argue that certain types of vertices must have been colored by Bob.) The case d = 0 can be proved in an easier way, if we use the (directed) (a, b)coloring number of a directed forest, which is ≤ b + 2 by a result in [2]. The proof of Theorem 2 consists in a similar idea. Here Alice’s invariant is the following: after her moves a connected region of uncolored vertices is neighbored with at most two colored vertices (instead of one). Therefore we need one more color than in Theorem 1. One should mention that, by the results of Kierstead [5], even in the most simple case d = 0 the class F of undirected forests has (a, b)-game chromatic number b + 2 if 2b ≤ a and (a, b) = (2, 1), and F has (a, b)-game chromatic number b + 3 if b ≤ a < 2b or (a, b) = (2, 1). This might indicate that the gap of 1 between lower bound and upper bound in Theorem 2 is a challenge for future research. References [1] Andres, S. D., Lightness of digraphs in surfaces and directed game chromatic number, submitted to: Discrete Math. [2] Andres, S. D., Asymmetric directed graph coloring games, submitted to: Discrete Math. [3] Bodlaender, H. L., On the complexity of some coloring games, Int. J. Found. Comput. Sci. 2, no.2 (1991), 133–147 [4] Chou, C.-Y., W. Wang, and X. Zhu, Relaxed game chromatic number of graphs, Discrete Math. 262 (2003), 89–98 [5] Kierstead, H. A., Asymmetric graph coloring games, J. Graph Theory 48 (2005), 169–185 [6] Neumann-Lara, V., The dichromatic number of a digraph, J. Combin. Theory B 33 (1982), 265–270 12 The Valve Location Problem Hans L. Bodlaender a, Alexander Grigoriev b, Nadejda V. Grigorieva c, Albert Hendriks a a Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, NL–3508 TB Utrecht, The Netherlands b Maastricht University, Quantitative Economics, P.O.Box 616, NL–6200 MD Maastricht, The Netherlands c Institute of Power Resources Transport (IPTER), 144/3, pr. Octyabrya, Ufa-450055, Russia Key words: Computational complexity, bounded treewidth, dynamic programming, binary search, valve location problem 1 Motivation A pipeline is the most economically efficient and environmentally friendly way to transport hazardous liquids and gases, e.g. crude oil or natural gas, over land. When operating normally, pipelines do not produce any pollution. However, due to external factors or pipe corrosion, accidents on pipelines sometimes happen and the accidental damage can be substantial. To control possible spills, every pipe system is usually equipped with special shutoff valves. Whenever the pipe system is depressurized, the valves automatically separate the pipe network into sections. Therefore, the quantity of hazardous liquid or gas potentially leaving the system is proportional to the total length of the pipes in the damaged section of the network separated by shutoff valves. In the application at hand, there is a given number of valves that can be placed. Thus, we want algorithms that solve the following problem. Given a pipe network and a number of valves to be installed, how to find a valve location that minimizes the maximal possible spill? 2 Graph theoretic formulation The problem can be formulated in graph theoretic terms in a natural way. Let G = (V, E) be an undirected graph representing a pipe network. Degree 1 13 vertices of the graph represent sources and destinations of the pipe network. Edges of the graph represent pipes. Let ωe ∈ Z+ denote the length of pipe e ∈ E. Vertices of degree 2 and higher represent connection points between the pipes. Let k be a number of valves to be installed. We assume that a valve can be located in any vertex v ∈ V . Consider a k-elementary separator V ′ ⊆ V partitioning set E into subsets E1 , E2 , . . . , ES where edges in Es , 1 ≤ s ≤ S, form a connected component in G, and for any two subsets, Es and Et , the set of endpoints in Es intersects the set of endpoints in Et only in elements of V ′ . We define Wmax (V ′ ) =  max1≤s≤S e∈Es ωe . The problem is to find a separator V ′ that minimizes Wmax (V ′ ). In other words, we have to find a k-elementary separator in G such that the maximum weight connected component is minimized. We refer to this problem as the valve location problem. We consider also the unweighted version of the problem where ωe = 1 for all e ∈ E. To the best of our knowledge this problem was not addressed in the literature before. A related problem is the problem to find small balanced separators in a graph, see e.g., [3]. 3 Simple networks: paths, cycles and trees For simple graphs, like paths and cycles, we can apply the textbook dynamic programming algorithm computing an optimal solution in O(kn2 log ωmax ) time, where n = |V | and ωmax = maxe∈E ωe . The recursive formulas in such dynamic programming are quite straightforward: for all 1 ≤ v ≤ n and 1 ≤ j ≤ k we have that f (v, j) = min max{f (u, j − 1), 1≤u≤v  ωw } (1) u≤w≤v−1 Using the specific of a min-max objective, we construct yet another algorithm that allows to obtain an optimal solution in O(n log ωmax log nωmax ) time. In this algorithm, given a positive integer C, we simply check whether k valves are sufficient to maintain the maximum inter-valve distance at most C or not. This checking can be done in O(n log ωmax ) time simply by going along the path (cycle) and consecutively placing the valves such that the distance to the valve-predecessor is maximized but not exceeding C. Then, binary search on C in interval [1, nωmax ] solves the valve location problem. Interestingly, the second algorithm can be extended to the valve location problem on trees. Here, in the checking subroutine, we move from leaves to a 14 root and at each step we separate by a valve the largest subtree or subforest of the total length at most C. 4 Graphs of bounded treewidth In practice, the long oil pipeline systems are more complicated than trees. This makes the problem more difficult from algorithmic perspective. Fortunately, most of the real-life pipe networks are outerplanar or, taking this more generally, the corresponding graphs have bounded treewidth. (See e.g., [1,2,4].) For this type of networks we have the following results. Theorem 1 The valve location problem on graphs of treewidth q admits a dynamic programming algorithm running in time (nωmax )O(q) . In this dynamic programming we move in down-up fashion from leaves to a root in the tree decomposition of G. In the tables, corresponding to a bag of the tree decomposition, we store the following information: (i) a number of valves installed in the branch below the bag; (ii) a valve location among the vertices belonging to the bag; (iii) connectivity of the vertices belonging to the bag; (iv) lengths of connected components containing the bag vertices; (v) maximum length connected component in the branch below the bag. Notice that this dynamic programming is only pseudo-polynomial time algorithm for the weighted version of the valve location problem on graphs of bounded treewidth. Using standard scaling arguments, we derive the following corollary. Corollary 2 The valve location problem on graphs of bounded treewidth admits an FPTAS. 5 General network structures For general network structures, we have the following complexity result. Theorem 3 The unweighted version of the valve location problem is NPhard. The proof of this theorem uses a reduction from the strong NP-hard problem 3-partition. Finally, we conjecture that the weighted version of the valve location problem is (weak) NP-hard even for graphs of bounded treewidth. 15 References [1] H.L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sc., 209:1-45, 1998. [2] H.L. Bodlaender. Treewidth: Characterizations, applications, and computations. In F. V. Fomin, editor, Proceedings 32nd International Workshop on Graph-Theoretic Concepts in Computer Science WG’06, pages 1-14. Springer Verlag, Lecture Notes in Computer Science, vol. 4271, 2006. [3] U. Feige and M. Mahdian. Finding small balanced separators. In Proceedings of the 37th Annual Symposium on Theory of Computing, STOC’06, pages 375384. ACM Press, 2006. [4] I.V. Hicks, A.M.C.A. Koster, and E. Kolotoglu. Branch and tree decomposition techniques for discrete optimization. In J.C. Smith, editor, TutORials 2005, INFORMS Tutorials in Operations Research Series, chapter 1, pages 1-29. INFORMS Annual Meeting, 2005. 16 Most balanced minimum cuts and partially ordered knapsack Paul S. Bonsma 1 Institut für Mathematik, Sekr. MA 6-1, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany Key words: balanced cuts, partially ordered knapsack, approximation algorithms 1 Introduction For a graph G = (V, E) and two non-empty, disjoint sets S ⊂ V and T ⊂ V , [S, T ] denotes the set of edges of G with one end vertex in S and one end vertex in T . Edge cuts or cuts are denoted as [S, S]. We study balanced cut problems that differ from the balanced cut problems that are usually studied (see e.g. [6]); instead of searching for a cut [S, S] with minimum number of edges among all cuts satisfying a balance requirement, we are looking for a most balanced cut among a set of cuts with requirements on the number of edges. For s, t ∈ V , a cut [S, S] is a minimum st-cut if s ∈ S, t ∈ S and it has minimum number of edges among all such cuts. MOST BALANCED MINIMUM st-CUT: (MBMC) INSTANCE: A graph G, two vertices s, t ∈ V (G). SOLUTION: A minimum st-cut [S, S]. GOAL: Maximize min{|S|, |S|}. This problem was previously studied by Feige and Mahdian [2], who also studied a vertex cut variant. They proved N P-hardness, and gave a fixed parameter tractable algorithm where the parameter is the number of edges resp. vertices in the cut. We show that this problem is closely related to partially ordered knapsack (POK), by giving a bijection between all minimum st-cuts and the ideals of Email address: bonsma @ math.tu-berlin.de (Paul S. Bonsma). Supported by the Graduate School “Methods for Discrete Structures” in Berlin, DFG grant GRK 1408 1 17 some partial order. From this idea a new N P-hardness proof follows. Our main result is a polynomial time approximation scheme (PTAS) for a special case of POK, which yields a PTAS for MBMC. For vertex cuts, there are multiple ways to define most balanced cuts. For one of these, the PTAS also works. As a motivation for this problem, Chimani, Gutwenger and Mutzel [1] give an integer program for calculating the crossing number of a graph G, and show that edge cuts [S, S] can be used in a preprocessing step to split the instance in two. This step is correct whenever [S, S] is a minimum st-cut for some pair s and t, and the gain is larger when the cut is more balanced. Another possible application of MBMC is as a subproblem in methods for important, but harder cut problems such as minimum α-balanced cut or sparsest cut (see [5] or [6]). The work of Feige and Mahdian [2] shows one example of such an application. Finally, the results contribute to the knowledge on approximating special cases of POK. Little is known about approximating POK in general (see [4] for other special cases). 2 Reducing MBMC to UPOK A partial order will be denoted as (B, ≺), where ≺ is an irreflexive, transitive relation on set B. I ⊆ B is an ideal of (B, ≺) if x ≺ y and y ∈ I implies x ∈ I. UNIFORM PARTIALLY-ORDERED KNAPSACK (UPOK) INSTANCE: A partial order ({1, . . . , k}, ≺), weights w : {1, . . . , k} → N, and an integer WU . SOLUTION: An ideal I with w(I) ≤ WU . GOAL: Maximize w(I). Consider a MBMC instance G, s, t. A critical edge of G is an edge that is part of at least one minimum st-cut. The critical edges can be found in polynomial time, by considering maximum st-flows (after setting all edge capacities to one, allowing flow in both directions), and checking for which edges a decrease in capacity yields a decrease in flow. Claim 1 A vertex partition {B1 , . . . , Bk } exists such that G[Bi ] is connected, and uv ∈ E(G) is critical if and only if u ∈ Bi and v ∈ Bj for some i = j. In addition, for every critical edge, in every maximum st-flow the flow goes in the same direction. Let i ≺ j if in a maximum st-flow, flow goes from a vertex in Bi to a vertex in Bj . Add transitive relations to ensure that ≺ is a partial order on {1, . . . , k}. Using these definitions, minimum st-cuts are characterized as follows: 18 Claim 2 [S, S] with s ∈ S is a minimum st-cut if and only if: • For all i: Bi ⊆ S, or Bi ⊆ S, and • if i ≺ j and Bj ⊆ S, then Bi ⊆ S. Even though the objectives are slightly different, the relation with UPOK now is obvious. Let P = ({1, . . . , k}, ≺), and in addition assign weights w(i) = |Bi |. Claim 3 G has a minimum st-cut [S, S] with s ∈ S, |S| = x if and only if P has an ideal I with weight w(I) = x. 3 A PTAS for a special case of UPOK and MBMC  We present a PTAS for UPOK instances with ki=1 w(i) ≤ CWU for some constant C. For a desired approximation ratio 1 − ǫ, define WL = (1 − ǫ)WU . Elements with w(i) > WU − WL are called large, other elements small. L denotes the set of large elements. For every L′ ⊆ L do If an ideal I exists with L′ ⊆ I and L\L′ ⊆ I then Let I be a minimum ideal with L′ ⊆ I. While a small element x exists such that I ∪ {x} is an ideal do: Add x to I. Return the best solution considered throughout the algorithm. The number of sets L′ considered is at most 2|L| ≤ 2C/ǫ , so the complexity of this algorithm is poly(k)2C/ǫ , which is polynomial for fixed C and ǫ. It can be argued that the algorithm will find an optimal solution, or a solution between WL and WU . Thus we have a (1 − ǫ)-approximation algorithm for every ǫ > 0. Combining this with the reduction from MBMC to UPOK, with proper choice of WU and a slightly different analysis, a PTAS is found for MBMC. Note that C < 2 in this case. Also note that considering ideals I with w(I) ≥ WU is only needed when using this algorithm for MBMC. 4 N P-completeness, reducing UPOK to MBMC In this section we consider the decision variants of both problems. For UPOK, in addition to the upper bound WU , a lower bound WL is given and the question becomes ‘is there an ideal I with WL ≤ w(I) ≤ WU ?’ Using similar ideas as before, UPOK instances can also be reduced to MBMC, proving the N P-completeness of the problem. 19 We first transform the UPOK instance as follows: add a unique minimum element x and maximum element y. Assign weights to x and y (and increase WL and WU by w(x) to maintain an equivalent instance), such that the total weight becomes WL + WU . Now for every element i we introduce a complete graph on kw(i) vertices. Choose s and t in the components corresponding to x resp. y. Edges can be added between these complete graphs such that these are exactly the critical edges with respect to s and t, and such that all minimum st-cuts correspond to ideals of the partial order, similar to Claim 3. Since UPOK is strongly N P-complete [3], we may assume the weights of the UPOK instance were encoded in unary. This makes it a polynomial transformation, to an equivalent MBMC instance. Theorem 4 The decision version of MBMC is N P-complete. We can also consider the general most balanced minimum st-cut problem (GMBMC), where the instance is just a graph G, and a feasible solution is a cut [S, S] that is a minimum xy-cut for some pair x, y (this is the problem most relevant for [1]). The above construction can be extended such that a most balanced minimum xy-cut for arbitrary x and y is always a minimum st-cut, for the chosen s and t. This shows that GMBMC is also N P-complete. 5 Most balanced vertex cuts Instances of Most Balanced Minimum Vertex Cut (MBMVC) consist again of a graph G and two vertices s and t. A solution is a minimum vertex cut C that separates s from t. Since a minimum vertex cut can result in multiple components, different objectives can be chosen. In [2], the objective of minimizing the size of the largest component is considered. Alternatively, one may wish to maximize the size of the smallest component. We consider a version closer related to the edge cut problem: MBMVC - 2-PARTITION INSTANCE: A graph G, two vertices s, t ∈ V (G). SOLUTION: A minimum st-vertex cut C and partition {S, T } of V (G)\C with s ∈ S, t ∈ T , [S, T ] = ∅. GOAL: Maximize min{|S|, |T |}. For this variant of MBMVC, we can also construct a partial order such that there is a bijection between ideals and minimum st-cuts similar to Claim 3. Hence the PTAS also works for this problem. However for the other variants we cannot do this, which leads to the question how well these variants can be approximated. For all variants, a construction similar to before proves N P-completeness. 20 References [1] M. Chimani, C. Gutwenger, and P. Mutzel. On the minimum cut of planarizations. Technical Report TR06-1-003, University of Dortmund, Germany, 2005. [2] U. Feige and M. Mahdian. Finding small balanced separators. In Proceedings of the 38th STOC, pages 375–384, 2006. [3] D. S. Johnson and K. A. Niemi. On knapsacks, partitions, and a new dynamic programming technique for trees. Math. Oper. Res., 8(1):1–14, 1983. [4] S. G. Kolliopoulos and G. Steiner. Partially-ordered knapsack and applications to scheduling. In Algorithms—ESA 2002, volume 2461 of Lecture Notes in Comput. Sci., pages 612–624. Springer, Berlin, 2002. [5] T. Leighton and S. Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787–832, 1999. Original result published in 1988. [6] D.B. Shmoys. Cut problems and their application to divide-and-conquer. In D.S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, 1997. 21 22 A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint Andrea Bettinelli Alberto Ceselli, Giovanni Righini Dipartimento di Tecnologie dell’Informazione Università degli Studi di Milano Via Bramante 65, 26013 Crema, Italy Abstract In this paper we consider a variation of the bin packing problem in which there are bins of different types with different costs and capacities. Furthermore, each bin has to be filled at least to a certain level, which depends on its type. We present a set partitioning formulation and an exact optimization algorithm which exploits column generation and specialized heuristics. We compare our algorithm with the general purpose solver ILOG CPLEX, running on two compact ILP formulations and we report on a preliminary evaluation on instances from the literature. Key words: Bin packing, column generation, branch-and-price. We consider the following variation of the bin packing problem: a set I of n items is given and each item i ∈ I has a given positive integer weight wi ; a set T of T bin types is also given; each bin of type t ∈ T has a capacity bt and a cost ct . Each bin of type t ∈ T also has a minimum filling constraint, that is it can be open only if the sum of the weights of the items assigned to the bin is greater than or equal to a given threshold at . The objective is to open a set of bins and to place each item into an open bin, minimizing the total cost of the open bins. The case in which at = 0 ∀t ∈ T is known as variable size bin packing problem (VSBPP). The literature on the VSBPP is quite rich: algorithms with an a priori guarantee of optimality have been presented in [1] and [2]. The cutting stock version of the VSBPP has been recently considered in [3] and [8]. A so called discretized formulation has been introduced in [4]. Email addresses: anbettinelli@crema.unimi.it (Andrea Bettinelli), {ceselli,righini}@dti.unimi.it (Alberto Ceselli, Giovanni Righini). 23 The minimum filling constraint complicates the problem from both a computational and a theoretical point of view, since even detecting the feasibility of an instance becomes an NP -hard problem, even in the case where an infinite number of bins of each type are available. A special case of the bin packing problem with minimum filling constraint ( BPPMFC ) is the bin packing problem with step cost function (BPPSCF), in which a1 = 0 and bt + 1 = at+1 ∀t = 1, . . . , T − 1; this problem arises when economies of scale are present in the transportation of packages and, at the best of our knowledge, it was first addressed by Walkowiak [9] who studied the optimal choice of mail packages in Poland. The author modeled the problem with a compact ILP formulation, and he solved real instances to optimality. A fast heuristic. Our heuristic is an adaptation of the well-known Best Fit heuristic for the BPP [5]. The items are considered in order of non-increasing weight. The bin types are considered in order of non-decreasing cost per unit of capacity ct /bt ; let bin types indices t1 , . . . , tT represent such an ordering. In each iteration i = 1, . . . , T the minimum filling constraint is initially neglected; the algorithm only considers bins of type ti and the Best Fit heuristic is executed: it assigns each item to a bin, but it may produce an infeasible solution, since some bin may not be filled enough to satisfy the minimum filling constraint. If this is the case, the underutilized bins are discarded, the items assigned to them remain unassigned, i is increased by 1 and the algorithm iterates with a new bin type. If there are still unassigned items after iteration T , then the heuristic fails. This routine is used both for obtaining feasible integer solutions and for tightening bounds in our algorithm. Set partitioning formulation. We model the BPPMFC as the following Set Partitioning Problem (SPP): minimize s.t.   t∈T k∈Ωt t∈T k∈Ωt   lt ≤ ct z k (1) xki z k = 1 ∀i ∈ I (2) z k ≤ ut ∀t ∈ T (3)  k∈Ωt L≤   t∈T zk ≤ U (4) k∈Ωt z k ∈ {0, 1} ∀t ∈ T , ∀k ∈ Ωt . (5) This formulation has a column for each feasible filling pattern of every bin. Each binary variable z k takes value 1 if pattern k is selected, 0 otherwise, and each binary coefficient xki is equal to 1 if and only if item i ∈ I appears in pattern k. Each set Ωt of indices of feasible patterns for bins of type t ∈ T  is defined as Ωt = {(xk1 . . . xkn ) ∈ {0, 1}n |at ≤ i∈I wixki ≤ bt }. The objective is to select a set of patterns of minimum cost; in order to satisfy partitioning constraints (2), each item has to be inserted in exactly one selected pattern. 24 Constraints (3) and (4) are only used to tighten the formulation in non-root nodes of the branch-and-price tree: the former set imposes lower (lt ) and upper (ut ) bounds on the number of selected patterns for each type, while the latter imposes lower (L) and upper (U) bounds on the total number of selected patterns. At the root node L = lt = 0 ∀t ∈ T and U = ut = n ∀t ∈ T . Branch-and-price. We use the continuous relaxation of model (1)-(5), the socalled master problem, to obtain a valid dual bound in a tree search algorithm. The number of elements in each set Ωt , and therefore the number of variables in our model, grows combinatorially with the number of items n. Hence, we adopt column generation. Column generation. Let πi ∈ R be the dual variables associated to each constraint (2), λt ≥ 0 and µt ≤ 0 be the dual variables associated to constraints (3), γ ≥ 0 and δ ≤ 0 be the dual variables associated to constraints (4). When an optimal solution for the linear relaxation of the restricted master problem (RLMP) is found, a search for new columns with negative reduced cost is performed by solving the following pricing subproblem for each bin type t ∈ T : minimize c̄k = ct − λt − µt − γ − δ − s.t. at ≤   πi xki i∈I wi xki ≤ bt i∈I xki ∈ {0, 1} ∀i ∈ I. The RLMP is enlarged with the column having the minimum negative reduced cost c̄k (if any) for each t ∈ T ; then the RLMP is solved again, obtaining a new set of dual variables and the column generation process is iterated. When the pricing problem does not identify any useful new column, the value of the optimal RLMP solution is a valid dual bound for the integer SPP. We perform heuristic pricing with a specialized heuristic; when the heuristic fails in finding new columns we switch to CPLEX to solve the pricing problem to optimality. When column generation is over, we try to refine both upper and lower bounds by solving a bounded knapsack problem. In order to obtain tighter dual bounds, we search for the best combination of bin types yielding a to∗ tal cost greater than or equal to the lower bound zRLM P provided by column generation. This requires to solve the following problem: minimize v =  ct y t t∈T s.t.  ∗ ct yt ≥ zRLM P t∈T yt ≤ Ut yt ∈ Z ∀t ∈ T ∀t ∈ T . 25 which can be tackled as a bounded knapsack problem, and efficiently solved ∗ by existing codes [6]. The value v replaces zRLM P as a final lower bound for the node of the search tree under examination. Furthermore, we try to accommodate the items in the set of open bins given by the optimal solution of the bounded knapsack problem above, by Best Fit. If a feasible solution is found with no underutilized bins, then the node can be fathomed, because we have found an upper bound an a lower bound that coincide. Otherwise we are let with a subset of unassigned items and we execute our heuristic on them, to look for a feasible solution hopefully better than the incumbent upper bound. Branching. We evaluated two branching schemes, described hereafter. • Branching on the number of bins. Let z̃ k be the (possibly fractional) value  of each variable z k in the optimal RLMP solution and let yt = k∈Ωt z̃ k be the (possibly fractional) number of open bins of type t in such solution. We select the bin type t̃ whose corresponding yt̃ variable has its fractional part closest to 0.5; then we perform binary branching, fixing lt̃ = ⌈yt̃ ⌉ in one branch and ut̃ = ⌊yt̃ ⌋ in the other branch.   • Branching on a pair of items. Let xij = t∈T k∈Ωt |xki =1∧xkj =1 z k represent how often items i and j occur in the same patterns in the fractional RLMP solution. As in the well-known Ryan-Foster rule, we select the pair of items (ı̃, ̃) whose corresponding xı̃̃ value is closest to 0.5 and we perform binary branching by introducing the constraint xı̃ + x̃ ≤ 1 in one branch and the constraint xı̃ = x̃ in the other branch. Computational results. Our algorithm has been implemented in C++ using SCIP [7]; we used CPLEX 8.1 both as an LP solver to optimize the RLMP and as an ILP solver to solve the pricing subproblem. We considered datasets for the VSBPP from the literature involving up to 500 items and 5 bin types, described in [1]. We fixed each at value as in the BPPSCF. We drew also a new dataset, setting each at to (bt − at )/2. We considered both a compact and a discretized formulation for the BPPMFC ; these correspond to the formulations for the VSBPP presented in [1] and [4]; in a preliminary experimental campaign we compared the bounds obtained using CPLEX on these formulations, with the SPP bound obtained by our algorithm. Even computing the continuous relaxation bound of the discretized formulation turned out to be a difficult problem: we could run experiments only on small instances, mainly due to memory overflow problems; on the opposite, the SPP bounds and the compact formulation bounds were always obtained in few seconds. The SPP bound dominated the compact formulation bound; moreover, on the largest instances both methods required about the same CPU time. Refining the SPP fractional solution by solving the bounded knapsack problem further improved the bound with negligible additional time. 26 References [1] Monaci M. (2002) Algorithms for Packing and Scheduling Problems, PhD thesis OR/02/4, DEIS - Università di Bologna [2] Alves C. and Valerio de Carvalho J.M. (2006) A stabilized branch-and-priceand-cut algorithm for the multiple length cutting stock problem, Computers and Operations Research, in press [3] Alves C. and Valerio de Carvalho J.M. (2006) Accelerating column generation for variable sized bin-packing problems, European Journal of Operational Research, in press [4] Correia, I., Gouveia L. and Saldanha da Gama F. (2006) Solving the variable size bin packing problem with discretized variables, Computers and Operations Research, in press [5] Martello S. and Toth P. (1990) Knapsack problems: algorithms and computer implementations, John Wiley and Sons – available online [6] Pisinger D. (2000) A minimal algorithm for the bounded knapsack problem, INFORMS Journal on Computing, 34:75?84 [7] Achterberg T. (2004) SCIP - a framework to integrate Constraint and Mixed Integer Programming, ZIB report 04-19, Berlin [8] Belov G. and Scheithauer G. (2002) A cutting plane algorithm for the onedimensional cutting stock problem with multiple stock lengths, European Journal of Operational Research, 141:274–294 [9] Walkowiak R. (2005) Shipment cost optimization, ECCO XVIII – Minsk 27 28 A latency problem with profits Sofie Coene and Frits C.R. Spieksma Katholieke Universiteit Leuven, Department of Operations Research, Naamsestraat 69, B-3000 Leuven, Belgium. Key words: minimum latency, traveling repairman, dynamic programming 1 Introduction Consider the following problem. Given are a set of clients located on the line and profits associated with each client. In addition, a single server is given, positioned at the origin at time t = 0. The server travels at unit speed. If the server visits client i at time t, the profit collected by the server equals pi − t (where pi denotes the profit associated with client i). The goal is to select clients and to find a route for the server visiting the selected clients, such that total collected profit is maximal. We refer to this problem as the traveling repairman problem with profits, or TRPP for short. Notice that in the TRPP (i) not every client needs to be visited, and (ii) the profit collected at a client depends on the time needed to reach that client. Our contribution in this note is modest: we formulate a dynamic program that solves the TRPP in polynomial time, thereby generalizing a classical result from Afrati et al. [1]. In Section 2 we refer to some literature and we motivate this problem. In Section 3 we describe the dynamic programming algorithm. 2 Literature and motivation The TRPP is a generalization of the well-known traveling repairman problem (TRP) on the line, also known as the minimum latency problem (MLT) on the line. No profits are given in this problem and the goal is to visit all clients with minimal total latency. Afrati et al. [1] give an O(n2 ) dynamic program which was later improved to O(n) by Garcı́a et al. [6]. Minieka [7] shows that this problem on a tree network is polynomial for trees with unit weights and develops a pseudo-polynomial time algorithm for the problem on weighted trees, which is proven to be NP-hard by Sitters [8]. The problem with multiple identical servers is solvable in O(n4), see Wu [9] and Averbakh and Berman [2]. For recent work on exact algorithms and approximation algorithms we refer 29 to Wu et al. [10], and Fackaroenphol et al. [4] respectively, and the references contained therein. In de Paepe et al. [3] a framework is described dealing with the computational complexity of dial-a-ride problems (which include latency problems, and in particular latency problems on the line). As far as we are aware there is no work on latency problems with profits. This is in contrast with the situation for the TSP, we refer to Feillet et al. [5] who survey the TSP with profits. Motivation The TRPP is interesting because (i) it is a first attempt to combine a latency objective with profits, (ii) the complexity of latency problems whose metric space is the line is often unresolved [3], (iii) TRPP occurs as the pricing problem of an integer programming formulation modeling the latency problem with multiple servers. Indeed, consider a situation with nonidentical servers (meaning that their speed may differ) whose job is to service a set of clients on the line. Then, using a set-partitioning formulation, it is the TRPP that arises as the pricing problem in a column generation approach. We do not go into further details here. 3 The algorithm We represent an instance of the TRPP on the line as depicted in Figure 1. The server starts in d = x0 = y0 . The clients x1 , x2 , . . . , xn and y1 , y2 , . . . , ym are to the left respectively to the right of the origin, and to each client a profit pxi resp. pyj (with i = 0, . . . , n; j = 0, . . . , m) is associated. The goal is to select clients and to find a route for the server such that total profits of the clients visited minus the latency of the corresponding route is maximal. Fig. 1. The TRPP on the line Observe that, for those clients that are visited, it is optimal to visit the client the first time the server passes by. The optimal tour thus has a spiral shape as illustrated in Figure 1 (see de Paepe et al. [3]). Given an instance of TRPP it is not yet clear which clients, and in particular how many clients, need to be selected. We deal with this issue by proposing a procedure that finds an optimal solution when selecting precisely K clients 30 (1 ≤ K ≤ n + m). Next, our algorithm consists in applying this procedure for each possible value of K. We now sketch our dynamic program (DP). A state in our DP is denoted by [xi , yj , l]K which corresponds to the situation where the server is positioned in xi (as its leftmost visited point), where the rightmost visited point is yj , where l clients (including xi and yj ) are visited within that interval, and where K clients will be visited (0 ≤ i ≤ n, 0 ≤ j ≤ m, 0 ≤ l ≤ K, 1 ≤ K ≤ n + m). Similarly, in the state [yj , xi , l]K the server is now at position yj , the leftmost visited point is xi , l clients are visited within this interval, and K clients will be served. We use the following definitions: Definition 1 • t[xi , yj ] is the distance between locations xi and yj • P [xi , yj , l]K equals the maximal value of the difference between (i) the profits of the l clients visited in [xi , yj ], and (ii) latency costs incurred for the first l clients taking into account that K clients will be visited. We refer to P [xi , yj , l]K as the net profit. • Li = {xi−1 , xi−2 , . . . , x0 } • Rj = {yj−1, yj−2, . . . , y0 } We compute the profit in a state as follows: P [x0 , y0, 0]K = P [y0 , x0 , 0]K = 0 for K = 1, . . . , n + m (1) For i = 0, . . . , n; j = 0, . . . , m; l = 0, . . . , K; K = 1, . . . , n + m: P [xi , yj , l]K = pxi + max{ P [z, yj , l − 1]K − (K + 1 − l) ∗ t[z, xi ], z∈Li P [yj , z, l − 1]K − (K + 1 − l) ∗ t[yj , xi ] } (2) P [yj , xi , l]K = pyj + max{ P [z, xi , l − 1]K − (K + 1 − l) ∗ t[z, yj ], z∈Rj P [xi , z, l − 1]K − (K + 1 − l) ∗ t[xi , yj ] } (3) In order to ensure correctness we assume that taking the maximum over an empty set equals −∞, thus P [x0 , yj , l]K = P [y0, xi , l]K = −∞ for i > 0 and j > 0. In addition, for states in which (i + j < l) or (l = 0 ∧ i + j > 0) or (l = 1 ∧ i ≥ 1 ∧ i ≥ 1) the net profit equals −∞. Then total profit for a given K is: T otalP rof itK = max{P [xi , yj , K]K } i,j 31 (4) and finally T otalP rof it = max{max{T otalP rof itK }, 0} K (5) Claim 1 Algorithm DP is correct. Proof. We state this claim without proof. ✷ To estimate the complexity of DP, let p = n + m, then for every K there are at most p3 states possible, every state has 2p elements in its maximization function, thus for every K the time complexity is O(p4 ). Total complexity of the algorithm is thus O(p5 ), and we can close by stating our result. Corollary 1 TRPP is solvable in polynomial time. References [1] Afrati F., S. Cosmadakis, C.H. Papadimitriou, G. Papageorgiou, and N. Papakostantinou (1986), The complexity of the traveling repairman problem, Informatique théorique et Applications 20, 1, 79-87. [2] Averbakh I. and O. Berman (1994), Routing and Location-Routing pDeliveryMen Problems on a Path, Transportation Science 28, 2, 162-166. [3] de Paepe W.E., J.K. Lenstra, J. Sgall, R.A. Sitters, and L. Stougie (2004), Computer-Aided Compexity Classification of Dial-a-Ride Problems, INFORMS Journal on Computing 16, 2, 120-132. [4] Fackaroenphol J., C. Harrelson, and S. Rao (2003), The k-Traveling Repairman Problem, Proceedings of the fourteenth annual ACM-SIAM Symposium on Discrete Algorithms. [5] Feillet D. ,P. Dejax, and M. Gendreau (2005), Traveling Salesman Problems with Profits, Transportation Science 39, 2, 188-205. [6] Garcı́a A., P. Jodrá, and J. Tejel (2002), A note on the traveling repairman problem, Networks 40, 1, 27-31. [7] Minieka E.(1989), The Delivery Man Problem on a Tree Network, Annals of Operations Research 18, 261-266. [8] Sitters R. (2002), The Minimum Latency Problem is NP-hard for Weighted Trees, Proceedings of the ninth International IPCO Conference on Integer Programming and Combinatorial Optimization. [9] Wu B.Y. (2000), Polynomial time algorithms for some minimum latency problems, Information Processing Letters 75, 225-229. [10] Wu B.Y., Z. Huang, and F. Zhan (2004), Exact algorithms for the minimum latency problem, Information Processing Letters 92, 303-309. 32 A two-phase greedy algorithm for modular lattice polyhedra Ulrich Faigle Zentrum für Angewandte Informatik (ZAIK), Universität zu Köln, D–50931 Köln, Germany Britta Peis Universität Dortmund, D–44227 Dortmund, Germany Key words: distributive lattice, submodular function, duality 1 Introduction In a series of papers, Hoffman [1], [2] and Hoffman and Schwartz [3] developed a theory of lattice polyhedra, which form a generalization of matroid polyhedra by allowing an order structure on the feasible sets that need not coincide with the “natural” set-theoretic ordering by containment: Let (L, ≤) be a finite (partially) ordered set. L is a pseudolattice if for all a, b ∈ L there are elements a ∧ b, a ∨ b such that a ∧ b ≤ a, b ≤ a ∨ b. A function r : L → R defined on a pseudolattice L = (L, ≤, ∧, ∨) is submodular if for all a, b ∈ L, r(a) + r(b) ≤ r(a ∧ b) + r(a ∨ b). The function r is supermodular if −r is submodular, and modular if r is both, sub- and supermodular. Let U be a (finite) set. A set representation of (L, ≤) is a map χ : L → 2U into the collection of subsets of U such that for all a, b, c ∈ L Email addresses: faigle@zpr.uni-koeln.de (Ulrich Faigle), bpeis@mathematik.uni-dortmund.de (Britta Peis). 33 (C0) a < b =⇒ χ(b) \ χ(a) = ∅. (C1) a ≤ b ≤ c =⇒ χ(a) ∩ χ(c) ⊆ χ(b). For any u ∈ U, we define the characteristic function χu : L → {0, 1} such that χu (a) = 1 if and only if u ∈ χ(a). Provided L is a pseudolattice, we call the representation χ sub- or supermodular if χu is sub-, respectively, supermodular for any u ∈ U. Given a pseudolattice L, a set representation χ : L → 2U and a function r : L → R, the polyhedron  Q ≡ {x ≥ 0 | xu ≤ r(a) for all a ∈ L}, or u∈χ(a) Q̄ ≡ {x ≥ 0 |  xu ≥ r(a) for all a ∈ L} u∈χ(a) is called lattice polyedron if χ is a supermodular representation and r is submodular, or, if χ is a submodular representation and r is supermodular, respectively. Hoffman and Schwartz [3] showed that lattice polyhedra are totally dual integral. Thus, for each integral weight function c : U → R+ , there exist integral optimal solutions of the primal-dual pair of linear programs (P ) (D) max{ x≥0  cu xu | u∈U y≥0 xu ≤ r(a) for all a ∈ L} and u∈χ(a)  min{   r(a)ya | a∈L ya ≥ cu for all u ∈ U}, χ(a)∋u if r is submodular and χ is supermodular, respectively, (P̄ ) (D̄) min{ x≥0  cu xu | u∈U max{ y≥0  xu ≥ r(a) for all a ∈ L} and u∈χ(a)  a∈L r(a)ya |  ya ≤ cu for all u ∈ U}, χ(a)∋u if r is supermodular and χ is submodular. As an application, min-max theorems with respect to (poly-)matroids, shortest paths in digraphs, (poly-)matroid intersection or chains and antichains in posets can be derived (see e.g. [4], pp. 1025-1028). For the case of polymatroid polyhedra, Edmonds [7] proved that the corresponding linear programming problems can be solved optimally with a primal and dual greedy algorithm. Slightly more general, if L is a distributive lattice with join-irreducible elments P , (i.e., those elements with exactly one lower neighbor), and χ : L → 2P is the “Birkhoff representation” such that 34 χ(a) = {p ∈ P | p ≤ a} ⊆ P, these problems can be solved with the known generalized (poly-)matroid greedy algorithms (see,e.g., [8, 9, 13, 11]). It appears difficult, however, to identify appropriate greedy algorithms for general lattice polyhedra. Frank [10] could provide a two-phase greedy algorithm to solve problems (P̄ ) and (D̄) if χ is supermodular and r is submodular, non-negative and monotone increasing. If χ is modular, Dietrich and Hoffman [6] have recently established a dual greedy algorithm for problem (D = ) min{ y≥0  a∈L r(a)ya |  ya = cu for all u ∈ U}, χ(a)∋u that returns an optimal solution of (D = ) in case such a solution exists at all and r is submodular. We extend Dietrich and Hoffman’s result and provide a two-phase greedy algorithm for (P ) and (D) in case χ is modular and r is submodular, non-negative and monotone increasing. We proceed as follows: In a first phase, with a dual greedy algorithm, we construct a feasible solution y of problem (D), whose non-zero values are attained along a chain in L. Since the algorithmic procedure of this algorithm goes back to Monge [12] we refer to it as Monge algorithm. It can be seen that this Monge algorithm corresponds to Dietrich and Hoffman’s algorithm in case (D = ) is feasible. In the second phase, we use the chain obtained by the Monge algorithm and construct in a greedy way a vector x : U → R such that x and y satisfy the complementary slackness conditions. We can prove that x is a feasible primal solution if r is submodular, non-negative and monotone increasing on L. It then follows from linear programming duality that the feasible solutions x and y are optimal. Finally, we prove that a pseudolattice with modular representation is in fact a distributive lattice. Using the Birkhoff representation, we construct a weight function on the join-irreducible elements of L such that we are able to actually solve (P ) and (D) with the known generalized (poly-)matroid greedy algorithms mentioned above. 2 The two-phase greedy algorithm We assume in the following always that the order (L, ≤) is a (pseudo)lattice with a modular representation χ relative to the ground set U. W.l.o.g. we assume χ(m0 ) = ∅ for the minimal element m0 ∈ L. 35 In the first phase of the greedy algorithm, the Monge algorithm computes a particular solution of problem (D) in a straightforward iterative procedure. To formulate it, we denote by ℓ(m) the set of all lower neighbors of m. The algorithm works as follows: (M1 ) Let m ∈ L be maximal and choose some lower neighbor m∗ ∈ ℓ(m) and u∗ ∈ χ(m) \ χ(m∗ ) such that c∗ = min max {cu | u ∈ χ(m) \ χ(m′ )} = cu∗ . ′ m ∈ℓ(m) (M2 ) Set ym = max{0, c∗ } and subtract ym from all cu with u ∈ χ(m). (M3 ) Replace L by L∗ = {a ∈ L | a ≤ m∗ }. (M4 ) Iterate until L = {m0 }. We refer to u∗ ∈ χ(m) as the representative of m with respect to the Monge algorithm. We call m active (in the Monge algorithm) if c∗ ≥ 0 is true and collect all active elements mj into the Monge chain M = {m1 < . . . < mk } ⊆ L. Furthermore, let π = u1 . . . uk be the corresponding representatives such that ui ∈ χ(mi ) \ χ(mi−1 ) for i = 1, . . . , k. By construction, the resulting vector y is  a feasible solution of (D) with the property χ(a)∋ui ya = c(ui ) for all ui ∈ π. In the second phase, we constuct a primal greedy vector xπ by modifying the components of the zero vector xπ = 0 iteratively as follows (G1 ) xπu1 = r(m1 ).  (G2 ) xπuj = r(mj ) − {xπui | i < j, ui ∈ χ(mj )} (j = 2, . . . , k).  Since u∈χ(mi ) xπu = r(mi ) holds for each active element mi , the feasible Monge vector y and xπ satisfy the complementary slackness conditions. It therefore remains to prove that xπ is a feasible solution of problem (P ) in case r is submodular, nonnegative and monotone increasing, to yield the following Theorem 2.1 If L is a pseudolattice with modular representation χ : L → 2U and r : L → R is submodular, nonnegative and monotone increasing, then the two-phase greedy algorithm constructs optimal solutions for (P ) and (D). 3 Distributivity Using the characterization of distributive lattices by exclusion of N5 - and M3 sublattices [5] we can observe 36 Theorem 3.1 If L is a pseudolattice with modular representation χ : L → 2U , then L is a distributive lattice. Let P denote the set of join-irreducible elements of L. Since L is distributive, each maximal chain a0 < . . . < an in L corresponds to a linear extension P = {p1 , . . . , pn }. Thus, given an arbitrary maximal chain and any u ∈ U, we can find unique elements pui , puj ∈ P such that u ∈ χ(ak ) iff i ≤ k and k ≤ j −1 by (C1). We are able to prove the relationship {a ∈ L | u ∈ χ(a)} ≡ {a ∈ L | a ≥ pui } \ {a ∈ L | a ≥ puj }, which allows us to construct a suitable weight function on P such that the known generalized polymatroid greedy algorithm can be applied to solve problems (P ) and (D). References [1] A. J. Hoffman: On Lattice Polyhedra II: Construction and Examples. Report RC 6268, IBM Research Division, San Jose, California, 1976. [2] A. J. Hoffman: On Lattice Polyhedra III: Blockers and anti-blockers of lattice clutters. Math. Programming Study 8 (1978), 197-207. [3] A.J. Hoffman, D.E. Schwartz: On partitions of a partially ordered set. Journal of Combinatorial Theory, Series B 23 (1977) 3-13. [4] A. Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. In: Algorithms and Combinatorics 24 (Graham, Korte, Lovász, Wigderson, Ziegler, eds.) Springer-Verlag Berlin Heidelberg (2003). [5] G. Birkhoff: Lattice Theory. American Mathematical Society 25, Providence, Rhode Island, (1967). [6] B.L. Dietrich and A.J. Hoffman: On greedy algorithms, partially ordered sets, and submodular functions. IBM J. Res. & Dev. 47 (2003), 25-30. [7] J. Edmonds: Submodular functions, matroids, and certain polyhedra, in: Combinatorial Structures and Their Applications, R. Guy et al. eds., Gordon and Breach, New York, 1970, 69-87. [8] U. Faigle and W. Kern: Submodular linear programs on forests. Math. Programming 72 (1996), 195-206. [9] U. Faigle and W. Kern: On the core of ordered submodular cost games. Math. Programming 87 (2000), 483-489. [10] A. Frank: Increasing the rooted-connectivity of a digraph by one. Math. Programming 84 (1999), 565-576. [11] U. Krüger: Structural aspects of ordered polymatroids. Discr. Appl. Math. 99 (2000), 125-148. [12] G. Monge, Déblai and Remblai. Mem. de l’Academie des Sciences, 1781. [13] M. Queyranne, F. Spieksma, and F. Tardella: A general class of greedily solvable linear programs. Math. Oper. Res. 23 (1998), 892-908. 37 38 A characterization for jump graphs containing complementary cycles Jinfeng Feng Lehrstuhl C für Mathematik, RWTH Aachen University 52056 Aachen, Germany Key words: Jump graph, Complementary cycles Abstract Let G = (V, E) be a graph and L(G) the line graph of G. The complement of L(G) is called the jump graph of G, denoted by J(G). Let Kr,s denote the complete bipartite graph with two partite sets containing r and s vertices, respectively. Harary and Nash-Williams [3] showed that the line graph L(G) of a graph G is Hamiltonian if and only if G = K1,n with some n ≥ 3 or G contains a dominating circuit. (A circuit C in G is dominating if every edge of G is incident with a vertex of C.) In 2004, Wu and Meng [5] characterized all jump graphs which are Hamiltonian. In 2005 , Liu [4] gave a characterization of pancyclic jump graphs. A graph G = (V, E) has complementary cycles if it contains two cycles C1 and C2 with V (C1 ) ∩ V (C2 ) = ∅ and V (C1 ) ∪ V (C2 ) = V (G). In this paper, we give a characterization for jump graphs containing complementary cycles. 1 Terminology and Introduction All graphs considered here are finite, undirected and simple. We refer to [1] for unexplained terminology and notation. Let G = (V (G), E(G)) be a graph. The symbols ∆(G) and δ(G) denote the maximum degree and the minimum degree, respectively. As usual, Pn , Cn and Kn are, respectively, the path, cycle and complete graph of order n. Kr,s is the complete bipartite graph with two partite sets containing r and s vertices. In particular, K1,s is called a star. Kn− denotes the graph resulting from Kn by deleting an edge while Kn+ the graph by adding a new vertex and joining 39 it to exactly one vertex of Kn . The graph K3 ◦ K1 is obtained from K3 by adding for every vertex x of K3 some new vertex yx only adjacent to x, which is sometimes called a net and denoted by N. We say two graphs G and H are disjoint if they have no vertex in common, and denote their union by G + H; it is called the disjoint union of G and H. The disjoint union of k copies of G is written as kG. The line graph L(G) of G has the edges of G as its vertices and two vertices of L(G) are adjacent if and only if they are adjacent in G. We call the complement of line graph L(G) the jump graph J(G) of G, i.e., the jump graph J(G) is the graph defined on E(G), and in which two vertices are adjacent if and only if they are not adjacent in G. Since both L(G) and J(G) are defined on the edge set of a graph G, and isolated vertices of G (if G has) play no role in line graph transformation and jump graph transformation, we assume that the graph G under consideration is nonempty and has no isolated vertices in what follows. Harary and Nash-Williams [3] showed that the line graph L(G) of a graph G is Hamiltonian if and only if G = K1,n , where n ≥ 3, or G contains a dominating circuit. (A circuit C in G is dominating if every edge of G is incident with a vertex of C.) In [2], Chartrand et al. presented some sufficient conditions for a jump graph to be Hamiltonian. In addition, they posed the following two conjectures. Conjecture A. Let G be a graph of order at least 7 and size q ≥ 5. If q ≥ 2∆(G), then J(G) is Hamiltonian. Conjecture B. Let G be a Hamiltonian graph of order n ≥ 7 and size q. If q ≥ 2n − 2, then J(G) is Hamiltonian. In [5], Wu and Meng presented the following. Theorem 1 (Wu and Meng [5]) Let G be a graph of size q ≥ 1. Then J(G) is not Hamiltonian if and only if G satisfies one of the following conditions: (1) q < 2∆(G). (2) q = 2∆(G) and G has an edge uv such that d(u) = d(v) = ∆(G). (3) G is isomorphic to K3 + P3 , K3 + 2K2 or C4 + K2 . 40 (4) G has a subgraph isomorphic to     K3+    K4       − if q = 6, or K3 ◦ K1 if q = 7, if q = 8. K4 (5) G is isomorphic to K5 . By Theorem 1, we see that J(G) is Hamiltonian if and only if G satisfies the following conditions: (1) 2∆(G) ≤ q. (2) d(u) + d(v) ≤ q − 1 for all uv ∈ E(G). (3) G is not isomorphic to any graph in Fig. 1. As a corollary, Conjecture B is confirmed, and Conjecture A is disproved. G5,1 G5,2 G6,4 G7,1 G7,6 G8,1 G5,3 G6,1 G7,2 G7,3 G7,7 G7,8 G8,2 G8,3 G6,2 G7,4 G6,3 G7,5 G7,9 G8,4 G7,10 K5 Fig. 1. H1 In [4], Liu gave a characterization of pancyclic jump graph. Theorem 2 (Liu [4]) J(G) is pancyclic if and only if G satisfies the following conditions: (1) 2∆(G) ≤ q. (2) d(u) + d(v) ≤ q − 1 for any two vertices u and v. (3) G is not isomorphic to any graph of H1 ∪ H2 (see Fig. 1, 2). 41 C5 K3 + K1,3 2K3 W5 K5 − K 2 Fig. 2. H2 In this paper, we consider the complementary cycles in jump graphs and characterize jump graphs containing complementary cycles. 2 Complementary Cycles in Jump Graphs Theorem 3 Let G be a graph of size q ≥ 6. J(G) has complementary cycles if and only if the following holds: (1) 2∆(G) ≤ q and (2) for q = 2∆(G), there is no edge uv with d(u) = d(v) = ∆(G). (3) for 6 ≤ q ≤ 9, G contains no subgraph isomorphic to some known graphs. References [1] J.A. Bondy, U.S.R. Murty. Graph Theory with Applications, Macmillan, London, 1976. [2] G. Chartrand, H. Hevia, E.B. Jarrett, M. Schultz. Subgraph distances in graphs defined by edge transfers, Discrete Math. 170 (1997), 63-79. [3] F. Harary, C.St.J.A. Nash-Williams. On eulerian and Hamiltonian graphs and line graphs, Canad. Math. Bull. 8 (1965), 701-710. [4] Jiping Liu. A characterization of pancyclic complements of line graphs, Discrete Math. 299 (2005), 172-179. [5] Baoyindureng Wu, Jixiang Meng. Hamiltonian jump graphs, Discrete Math. 289 (2004), 95-106. 42 Dynamic Programming for Queen Domination Henning Fernau a a Universität Trier FB 4—Abteilung Informatik, 54286 Trier, Germany Key words: chessboard problems, queen domination, 1 Introduction Queen domination is one of the typical programming exercises of a first year’s computer science course. However, little work has been published on the complexity of this problem. We will discuss three approaches in the following sections: — backtracking, —dynamic programming on subsets, —dynamic programming based on path decomposition. Finally, we briefly discuss some complexity issues regarding this problem. One instance is the so-called Five Queens Problem on the chessboard: it is required to place five queens on the board in such positions that they dominate each square. This task corresponds to dominating set as follows: the squares are the vertices of a graph; there is an edge between two such vertices x, y iff a queen placed on one square that corresponds to x can directly move to y (assuming no other pieces on the board). Hence, the edge relation models the way a queen may move on a chess board, yielding the queen chessboard graph. According to [6], the general version played on a (general) n × n board is also known as the Queen Domination Problem, or Board Covering Problem [11]. The task is to find the minimum number of queens that can be placed on a general chessboard so that each square contains a queen or is attacked by one. Recent bounds on the corresponding domination number can be found in [4,5,7,10]. It appears to be that the queen domination number is “approximately” n/2, but only few exact values have been found up to today. By way of contrast, observe that on the n × n square beehive, the queen domination number has been established to be ⌊(2n + 1)/3⌋, see [13]. 43 Let us mention that, although the structure we started with, namely the chessboard, is of course a planar structure, the queen chessboard graph is not planar: in particular, all vertices that correspond to one line on the chessboard form a clique, i.e., the family of queen chessboard graphs contains arbitrary cliques, quite impossible for planar graphs by Kuratowski’s theorem. A nice treatment of both the independence and the domination problem can be also found in http://mathworld.wolfram.com/QueensProblem.html. 2 A solution by backtracking This is most likely to be the outcome of a typical programming exercise based on this problem. As can be easily seen, the complexity of this approach is O(n !) which is at least theoretically worse than the solutions developed belows. We are therefore only brief in this section. Can we do better ? In practice, for sure. The techniques described by Östergård and Weakley [10] obviously provide tremendous speed-ups. However, we are not aware of any proof that shows that, speaking in asymptotic terms, their algorithm runs faster than O(cn log n ) for some constant c. Their technique is basically relying on a cute pointer management in backtracking algorithms. 3 Dynamic programming We can get better algorithms by using more advanced algorithm techniques. This also proves that Queen domination can be not only used to teach backtracking, but may be also employed for exemplifying other techniques, as well. The main observation for the dynamic programming approach is the following one: for each set of lines (be them horizontal, vertical, or diagonal), the information how many queens are sufficient to dominate those lines is basically sufficient to gradually compute the minimum number of queens required to dominate an n × n board. Since there are n horizontal, n vertical, and 4n − 2 diagonal lines, in total subsets of less than 6n lines have to be maintained, leading towards an O(cn ) algorithm for Queen domination for some c. However, since different sets of dominated lines can lead to the same set of dominated squares, we must do our bookkeeping in terms of those dominated squares. Naively, this would mean that we maintain an array indexed by subsets of dominated squares where we store the number of queens needed to dominated exactly those squares. However, this naive approach would again 2 lead to resource requirements that are of order 2n . Conversely, consider the 44 Algorithm 1 A dynamic programming algorithm for Queen domination Require: positive integer n Ensure: minimum number of queens necessary to dominate all squares on an n × n board Set Q to {(∅, 0)}. {Q should be organized as a directed acyclic graph GQ reflecting the inclusion relation of the first components; the “root” of GQ is always (∅, 0). A directed spanning tree of GQ is used to investigate GQ in the second FOR-loop.} for all squares s of the chessboard do Let T be the set of (at most six) lines that contain s. for all line sets S do {Start the search through GQ at the root} Look for some (g(S), i) in Q using GQ . if NOT g(T ) ⊆ g(S) then {putting a queen on s would dominate new squares} Look for some (g(S ∪ T ), j) in Q using GQ . if found then Replace (g(S ∪ T ), j) by (g(S ∪ T ), min{j, i + 1}) else Add (g(S ∪ T ), i + 1) to Q (maintaining GQ ) Let M be the set of all possible lines. Find (g(M), j) in Q. return j mapping g that associates to a set of lines the corresponding set of squares,  i.e., g(S) = ℓ∈S ℓ, where S is a set of lines ℓ and a line ℓ is viewed as a set of squares. Since there are at most cn sets of lines, there are also at most cn sets of squares in the range of g. Then, we run Alg. 1. Finally, we can look up the desired result at g(M), where M is the set of all lines. In Alg. 1, we maintain a queue that contains information in the form of pairs (g(S), i) where partial solutions are encoded. More specifically, S is a set of already dominated lines, and i shows how many queens are needed to exactly dominate the quares in the lines of S. This semantics justifies the initialization: the empty set is dominated by zero queens. Of course, one could also in addition store information about the partial solution (dominating set) that attains the value i. What is the running time of Alg. 1? All operations but the loop through all line subsets take polynomial time. Theorem 1 Queen domination can be computed in time O(64n p(n)) when using Alg. 1, where p is a polynomial. Can we further improve on this algorithm that is based on dynamic programming on subsets ? The ideas presented in [12] seem not to apply. 45 4 Treewidth technique A graph of small treewidth, or from an algorithmic point even more interesting, of small pathwidth, is appealing since it allows different but standard forms of dynamic programming techniques to be applied, often generalizing algorithms as known on trees or paths. This notion can be formalized as follows. In what follows, we start with the n × n planar grid with diagonals that corresponds to a chessboard in a straightforward interpretation. This graph has pathwidth n + 1. For our dynamic programming algorithm based on the described path decomposition (solving the problem in time O(cn ) for a c to be determined), we endow each vertex of this graph (i.e., each square on the chessboard) within the bag with the following information: for each direction (queen movement) of that specific square we have to keep track of how it might still “influence” the yet non-examined squares. Naively, this is done as follows: To each vertex of the bag, we associate an 9-bit vector b = (bւ , b← , bտ , b↑ , bր , b→ , bց , b↓ , q). Consider now, w.lo.g., a bag in the ↓-phase and a vertex in the first column of that bag. Then, the first four components describe the possible situations according to the “past” of the computation of the graph, i.e., whether the square (represented by the vertex) is known to be dominated by some queen from the south-west (then, bւ = 1), from the west b← , from the north-west bտ , or from above b↑ . The next four components describe something about the “future” of the computation, e.g., b→ = 1 iff we expect that this square will be dominated from the right. q = 1 means that a queen is placed on that particular square. Hence, to a specific bag we can associate a table of size 29(n+2) . Each table entry indicates the minimum number of queens that is necessary to dominate the queen graph processed so far, assuming the specific bit vector settings. This means, in particular, that there need not be a queen on any line that has been partially processed so far (say, a particular row r) if we consider the case when the according bit vector(s) associated to the square(s) of the current bag in row r show a one in bit b→ . We can assume (and check if necessary) consistency of the table, marking inconsistent situations with ∞ as a table entry. For example, if there are two squares x and x′ in the bag such that x′ is the right neighbor of x, then bit vector b of x with b← = 1 can only be consistent with bit vectors b′ of x′ where b′← = 1. Conversely, b′← = 1 is only possible if q = 1 or if b← = 1. There are also some clearly inconsistent situations from the geometry of the chessboard. For example, a square in the upper row cannot be dominated from any northern direction. How to update the information can be best described when assuming that we 46 intercalate one more bag Bn′ in the path decomposition between Bn and Bn+1 by setting Bn′ = Bn \ {w} in the notation described above. Then, table t′n (corresponding to Bn′ ) is obtained from tn (associated to Bn ) by computing, for each (n + 1) × 9-bitvector b = (b1 , . . . , bn+1 ), t′n (b) = min9 tn ((b, b1 , . . . , bn+1 )). b∈2 Here, we assume (w.l.o.g.) that the bit vector of w is occupying the first 9 bits of the bit vector that is used to index tn . When computing tn+1 (associated to Bn+1 ) from t′n , we will always check for consistency. Hence, we can assume consistency of all tables by induction, based on starting with a consistent table for the first bag. Again, we refrain from giving all details here, but only mention that in the case when we assume q = 1 in the bit vector of the new vertex, i.e., when we put a queen on that square, then we are consistent only with those (n + 1) × 9-bitvectors where we have b→ = 1 for the 9-bitvector corresponding to the left neighboring square; similar conditions apply for other neighbors. Do we actually need all those bits as described above ? This is quite an important issue, since reducing bits means that we might be able to reduce the basis of the exponential function cn that basically determines the running time of the sketched path-decomposition based algorithm. We are currently at c = 29 and hence we get an algorithm that is worse than our previous one. However, we can “compress” those bits considerably, finally obtaining: Theorem 2 Queen domination can be solved in time O(15n n2 ), given an n × n chessboard. Further details on pathwidth and treewidth and according algorithms can be found in [1,3,2,8,9]. We also mention that it is rather easy to modify our sketched path decomposition based algorithm (by dropping the d-bit) to obtain an O(8n n2 ) algorithm to compute the number of optimum solutions to the queen independence problem, matching the running time of the algorithm given in [12] that is based on dynamic programming on subsets. 5 Complexity issues Surprisingly little is known regarding the computational complexity of Queen domination; for example, it is unknown if the natural decision problem is N P-hard, given n in unary. One can argue (in view of the relatively tight combinatorial bounds) that this is unlikely to achieve; however, it seems to be also difficult to produce a polynomial-time algorithm for this problem, keeping 47 in mind our struggle to obtain single-exponential algorithms for the problem (measured against the side length of the board). This complexity question is somewhat embarassing from a pedagogical point of view: if we teach both basics of programming and of complexity theory at undergraduate level, it is nagging that an answer is still pending to the seemingly simple question if a polynomial-time algorithm is achievable. After all, this might mean that this favorite backtracking example would need no backtracking at all. References [1] S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12:308–340, 1991. [2] H. Bodlaender. Discovering treewidth. In P. Vojtáš, M. Bieliková, B. CharronBost, and O. Sýkora, editors, SOFSEM, volume 3381 of LNCS, pages 1–16. Springer, 2005. [3] H. L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1–45, 1998. [4] A. P. Burger and C. M. Mynhardt. An upper bound for the minimum number of queens covering the n×n chessboard. Discrete Applied Mathematics, 121:51–60, 2002. [5] A. P. Burger and C. M. Mynhardt. An improved upper bound for queens domination numbers. Discrete Mathematics, 266:119–131, 2003. [6] M. J. Chlond. IP modeling of chessboard placements and related puzzles. INFORMS Transactions on Education, 2(2):56–57, 2002. [7] E. J. Cockayne. 86:13–20, 1990. Chessboard domination problems. Discrete Mathematics, [8] H. Fernau. Parameterized Algorithmics: A Graph-Theoretic Approach. Habilitationsschrift, Universität Tübingen, Germany, 2005. [9] T. Kloks. Treewidth. Computations and Approximations, volume 842 of LNCS. Springer, 1994. [10] P. R. J. Östergård and W. D. Weakley. Values of domination numbers of the queen’s graph. Electronic J. Combin., 8:#R29, 19pp., 2001. [11] V. Raghavan and S. M. Venkatesan. Bounds for a board covering problem. Information Processing Letters, 25:281–284, 1987. [12] I. Rivin and R. Zabih. A dynamic programming solution to the n-queens problem. Information Processing Letters, 41:253–256, 1992. [13] W. F. D. Theron and G. Geldenhuys. Domination by queens on a square beehive. Discrete Mathematics, 178:213–220, 1998. 48 Stochastic Programs with Dominance Constraints Induced by Mixed-Integer Linear Recourse Ralf Gollmer, Uwe Gotzes a, Frederike Neise, Rüdiger Schultz a gotzes@math.uni-duisburg.de Key words: Stochastic integer programming, stochastic dominance, mixed-integer optimization Stochastic programming models are derived from random optimization problems with information constraints. We start out from the following random mixed-integer linear program. min{c⊤ x + q ⊤ y : T x + W y = z(ω), x ∈ X, y ∈ Y }, (1) together with the information constraint that x must be selected without anticipation of z(ω). This leads to a two-stage scheme of alternating decision and observation: The decision on x is followed by observing z(ω) and then y is taken, thus depending on x and z(ω). Accordingly, x and y are called first- and second-stage decisions, respectively. X and Y are polyhedra, possibly involving integer requirements to some vector components. z is a random vector on some probability space. Furthermore - besides other implicit regularity assumptions - all objects in (1) may have conformal dimensions. The mentioned two-stage dynamics becomes explicit by the following reformulation of (1)  min c⊤ x + min{q ⊤ y : W y = z(ω) − T x, y ∈ Y } : x ∈ X x y = min {c⊤ x + Φ(z(ω) − T x) : x ∈ X} x  (2) where Φ(t) := min{q ⊤ y : W y = t, y ∈ Y }. 49 (3) The function Φ, called the value function of the mixed-integer linear program min{q ⊤ y : W y = t, y ∈ Y }, has been studied in parametric optimization. In view of (2), the random optimization problem (1) gives rise to the family of random variables  c⊤ x + Φ(z(ω) − T x)  x∈X (4) . Thus every first-stage decision x ∈ X induces a random variable fx (ω) := c⊤ x + Φ(z(ω) − T x). Traditional two-stage stochastic programming aims at optimizing nonanticipative decisions, i.e., finding a “best” x, or in other words a “best” member in the family (4) of random variables. For the specification of “best”, statistical parameters reflecting mean and/or risk are used. Employing the weighted sum of E and some risk measure R leads to mean-risk models min{E(fx ) + ρ · R(fx ) : x ∈ X} (ρ ≥ 0 fixed). (5) Here, we take an alternative view. Rather than heading for “best” members of (4), we want to identify “acceptable” members, and optimize over them. This leads to a new class of stochastic integer programs, see (6) below, whose algorithmic treatment is the focus of our presentation. Stochastic dominance, an established concept in decision theory, provides a possibility to formalize the above mentioned “acceptability”. We deal with first- and second-order stochastic dominance. When preferring small outcomes to big ones, a random variable X is said to dominate a random variable Y to first (second) order (X 1(2) Y) iff E(h ◦ X) ≤ E(h ◦ Y) for all nondecreasing (convex) functions h for which both expectations exist. In other words a random variable X dominates a random variable Y to first order stochastically, iff X attains lower values with higher probability than Y. Second-order dominance holds, iff X exceeds each value t to lesser extent than Y for each t ∈ R. Coming back to our two-stage random optimization problem (1) and the related family (4), we assume some (random) benchmark cost profile d be given. We consider only those x ∈ X “acceptable” for which the corresponding fx dominates the benchmark profile d to first or second order, respectively. Over all “acceptable” x ∈ X we optimize some function g : X → R. This leads to the following stochastic program with dominance constraint induced by mixed-integer linear recourse min{g(x) : fx i d, x ∈ X} , i = 1, 2 50 (6) In [1] and [2] the authors have shown that (6) can be approximated through discrete approximations of fx and d. On the other hand (6) can be shown to be equivalent to deterministic block-structured large-scale mixed-integer linear programs in the case of finitely many realizations of fx and d (cf. [1], [2]). However these so called deterministic equivalents quickly become extraordinary large and cannot be tackled with standard solvers alone. We discuss a decompositon approach which we exploit in a branch and bound scheme to solve (6). Amongst others we outline Lagrangean Relaxation as a lower bounding procedure. Upper bounds are established via several tailored feasibility heuristics. We conclude our talk by presenting successful applications of our implementation of the discussed techniques. For first order dominance instances we draw on examples from the field of energy investment. Second order instances are built on a distributed generation system. References [1] Gollmer, R.; Neise, F.; Schultz, R.: Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourse, Preprint 6412006, Department of Mathematics, University of Duisburg-Essen, 2006. [2] Gollmer, R.; Gotzes, U.; Schultz, R.: Second-order stochastic dominance constraints induced by mixed-integer linear recourse, Preprint 644-2007, Department of Mathematics, University of Duisburg-Essen, 2007. . . . and the references therein. The above Preprints can be accessed via: http://www.uni-duisburg.de/FB11/disma/preprints.shtml 51 52 Edge intersection graphs of single bend paths on a grid Martin Charles Golumbic1, Marina Lipshteyn1, Michal Stern1,2 1 Caesarea Rothschild Institute, University of Haifa, Israel, 2 The Academic College of Tel-Aviv - Yaffo, Israel 1 Introduction We combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let P be a collection of nontrivial simple paths on a grid G. We define the edge intersection graph EP G(P) of P to have vertices which correspond to the members of P, such that two vertices are adjacent in EP G(P) if the corresponding paths in P share an edge in G. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EP G(P) for some P and G, and < P,G > is an EPG representation of G. Theorem 1.1 Any graph is an EPG graph. In the proof of the theorem we construct an EPG representation for any graph. A turn of a path at a grid point is called a bend. An EPG representation is Bk -EPG if each path has at most k bends. A graph that has a Bk -EPG representation is called Bk -EPG. We consider here EPG representations in which every path has at most a single bend, called B1 -EPG representations and the corresponding graphs are called B1 -EPG graphs. Theorem 1.2 Any tree is a B1 -EPG graph. In the proof of the theorem we construct an B1 -EPG representation for any tree. 53 (a) (b) Figure 1. (a)The claw clique (b)The edge clique 2 Characterization of cliques and 4-cycles in B1 -EPG graphs The claw graph K1,3 consists of one central vertex and three independent vertices that are adjacent to the central vertex. Let < P,G > be a B1 -EPG representation of G. For any grid edge e in G, the collection {P ∈ P|e ∈ P } corresponds to a clique in G and is called an edge clique. For any copy of the claw in G, the collection {P ∈ P|P contains two edges of the claw} corresponds to a clique in G and is called a claw clique. See Figure 2. Theorem 2.1 Let < P,G > be a B1 -EPG representation of a graph G. Every clique in G corresponds to either an edge-clique or a claw-clique in < P,G > . Let < P,G > be a B1 -EPG representation of a graph G. Consider a 4-star subgraph of G with a center grid point b and the grid edges (a1 , b), (a2 , b), (a3 , b), (a4 , b) in clockwise order. A true pie is a 4-star such that each “slice” (ai , b) ∪ (ai+1 , b) for i = 1, . . . , 4 is contained in a different member of P, where addition is assumed to be modulo 4. In a true pie, each one of the four paths has a single bend. A false pie is a star such that each “slice” (a1 , b) ∪ (a2 , b), (a2 , b) ∪ (a4 , b), (a3 , b) ∪ (a4 , b), (a3 , b) ∪ (a1 , b) is contained in a different member of P. In a false pie, two of the paths (that correspond to non-adjacent vertices) have a single bend and the other two have no bends. Consider a rectangle subgraph of G of any size with four corners (x1 , y1 ), (x2 , y1 ), (x2 , y2), (x1 , y2 ). A frame is a rectangle such that there exist four paths P1 , . . . , P4 where each corner is a bend for different members among P1 , . . . , P4 , the subpaths P1 ∩ P2 , P2 ∩ P3 , P3 ∩ P4 , P4 ∩ P1 share at least one edge and the subpuths P2 ∩ P4 , P1 ∩ P3 do not share an edge. See Figure 2. Theorem 2.2 Let < P,G > be a B1 EPG representation of a graph G. Every 4-cycle in G corresponds to either a true pie or a false pie or a frame in < P,G > . 54 (a) (b) (c) Figure 2. (a)The true pie (b)The false pie (c)The frame Figure 3. The graph S4 . 3 Structural properties of B1 -EPG graphs We give a structural property that enables to generate non B1 -EPG graphs. Consequently, some of well-known graphs appear to be non B1 -EPG graphs. Let C be any subset of the vertices of a graph G. The branch graph B(G/C) of G over C has a vertex set, V (B), consisting of all the vertices of G not in C  but adjacent to some member of C, i.e., V (B) = v∈C {Adj(v)−C}. Adjacency in B(G/C) is defined as follows: we join x and y by an edge in E(B) if and only if in G, (1) x and y are not adjacent, (2) x and y have a common neighbor u ∈ C, (3) the sets Adj(x) ∩ C and Adj(y) ∩ C are not comparable, i.e., there exist w, z ∈ C such that w is adjacent to x but not to y, and z is adjacent to y but not to x. Theorem 3.1 Let < P,G > be a B1 -EPG representation of G, and let C be a maximal clique of G. If C corresponds to an edge clique in < P,G > , then the branch graph B(G/C) can be 2-colored. If C corresponds to a claw clique in < P,G > , then the branch graph B(G/C) can be 3-colored. The graph Si consists of a clique of size i on the vertices v1 , . . . , vi , and the vertices vi+1 , . . . , v2i , where vi+j is adjacent only to the vertices vj and vj+1 , and v2i is adjacent to vi and v1 . See Figure 3. Corollary 3.2 The B1 -EPG graphs are Si -free for i > 3. 55 56 Optimal Bundle Pricing for Homogeneous Items Alexander Grigoriev a, Joyce van Loon a, Maxim Sviridenko b, Marc Uetz a, Tjark Vredeveld a a Maastricht University, Quantitative Economics, P.O.Box 616, NL–6200 MD Maastricht, The Netherlands b IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heigths, NY 10598, USA Key words: Algorithm design, computational complexity, approximation algorithms, price optimization 1 Introduction Consider the situation that we want to sell a set of homogeneous items I = {1, . . . , m} to a set of bidders J = {1, . . . , n}, each of which is interested in exactly one subset, or bundle, of items Ij ⊆ I. The fact that each bidder is only interested in one particular bundle is referred to as single mindedness [7]. We have a certain amount of copies ci of each item i ∈ I available, and this amount may be limited or unlimited (i.e. ci ≥ n for all i ∈ I), as in the case of non-digital or digital goods, for example. Bidder j’s valuation bj ≥ 1 determines the maximum amount that she is willing to pay for her bundle Ij . We assume that each bidder’s valuation is known to the seller. A bidder is a winner if she gets assigned her bundle (at an affordable price), and a loser otherwise. The set of winners is denoted by W ⊆ J. A solution to the problem is a price p(j) that bidder j ∈ J has to pay for her bundle Ij . Later we will be more specific about further restrictions on the prices. A solution is called feasible if all winners can afford their respective bundles, and if no item is oversold. In the setting with single minded bidders considered here, a solution is envy-free if in addition, for all losers the respective bundle is priced higher than their valuation. The problem that is mainly addressed in the literature is the one with unlimited availability of items, the requirement that the solution is envy-free [1–6], and combinatorial pricing, where each item is assigned a price, and bundle prices are defined by the sum of the respective item prices. For this problem 57 the maximum revenue is hard to approximate to within a semi-logarithmic factor in the number of bidders [4]. In particular, it is unlikely that a constant approximation algorithm exists. For the same problem, Hartline and Koltun [6] present an approximation scheme with almost linear running time, given that the number of distinct items is constant. Moreover, Balcan and Blum [2] derive an O(k)-approximation, given that each bidder is interested in bundles of at most k items. Finally, there exist two fully polynomial time approximation schemes [2,3] for the problem where the bidders’ bundles are nested, that is, for any two bundles Ij and Ij ′ it holds that Ij ⊆ Ij ′ , Ij ′ ⊆ Ij or Ij ∩ Ij ′ = ∅. 2 Combinatorial pricing with global envy-freeness In combinatorial pricing, we need to assign an (anonymous) item price for each of the items, and a bidder j’s bundle is priced at the sum of the prices of items in the bundle. To be in line with previous papers on the same topic, from now on let us write pi for the price of item i, and the price of a bundle  I ′ ⊆ I is p(I ′ ) = i∈I ′ pi . As before, the item prices need to yield a feasible and envy-free solution, and we wish to maximize the total revenue, which can   be written as j∈W i∈Ij pi . Notice that in case of unlimited availability of items both feasibility and envy-freeness is in fact no issue – yet finding optimal prices is hard to approximate to within a semi-logarithmic factor [4]. For this reason we introduce a new condition, namely global envy-freeness, which is inspired by so called price ladder constraints that have been proposed before in slightly different settings [1,8]. We require that p(I ′ ) ≤ p(I ′′ ) if |I ′ | < |I ′′ |, for any two subsets of items I ′ and I ′′ , which assures that larger bundles are at least as expensive as smaller bundles. This is a strong condition, as it implies that most item prices are of roughly the same order of magnitude. Nevertheless the problem is non-trivial. Theorem 1 The revenue maximization problem with combinatorial pricing and unlimited availability of items is strongly NP-hard under the global envyfreeness condition. We show that the problem with global envy-freeness admits a PTAS, thus breaking the semi-logarithmic inapproximability barrier for the general case. Theorem 2 The pricing problem with global envy-freeness condition admits a PTAS. To derive this PTAS for the problem with combinatorial pricing and global envy-freeness, we restrict the prices to powers of (1 + δ) for some δ > 0. Assume, without loss of generality, that p1 ≤ p2 ≤ . . . ≤ pm , then by global 58   envy-freeness, we know k pk ≥ (k − 1)pm−k+2 for k = 2, . . . , m2 . The idea for the PTAS is now the following. Except for a constant number of the cheapest and most expensive items, all items have prices in roughly the same range. Therefore we can price all except a constant number of items uniformly with the same price, without loosing too much in terms of the total revenue. We therefore enumerate over all possible uniform prices for the bulk of the items, and over all possible combinations of prices for the remaining (constant number of) items. For the problem with limited availability of items, we simultaneously check feasibility and envy-freeness. 3 Pricing with affine price functions In this section, we consider a more general pricing model. Here, the price of bidder j’s bundle is determined by an affine function, p(j) = aj0 +aj1x1 +. . .+ ajK xK . The coefficients ajk , k = 1, . . . , K, are arbitrary coefficients that are given for all bidders j ∈ J. These coefficients may, in general, depend on both the bundle Ij and bidder j itself. Thus it may be the case that two bidders with the same bundle pay different prices. The pricing problem consists of determining values for the variables xk , k = 1, . . . , K. To stress the generality of this definition, let us give a specific example. If we let K = 1 and define aj1 = |Ij | for all bidders j, the bundle prices are determined by affine functions that depend only on the size of the bundles. The optimization problem is to determine the per-item price x1 . The combinatorial pricing problem is a special case of the pricing problem with affine price function. Let K = m, and for all bidders j ∈ J and all items i ∈ I, let aji = 1 if item i ∈ Ij , and aji = 0 otherwise. Then, xi can be interpreted as the price of item i, and the price of any bundle Ij equals the sum of its  item prices, i∈Ij xi . The optimization problem is to determine optimal item prices. In the full version of this paper, we describe a simple algorithm that solves this problem in polynomial time, as long as the dimension K of the affine price functions is constant. This algorithm is based on a polyhedral characterization of an optimum solution. Theorem 3 The revenue maximization problem with affine price functions can be solved in polynomial time, for constant K. The running time of the algorithm is exponential in K. If the dimension K of the price functions is not constant, the optimum revenue is hard to approximate to within a semi-logarithmic factor as this problem contains the combinatorial pricing model as a special case. Therefore, the hardness result 59 follows immediately from [4]. On the negative side, it turns out that the problem with limited availability of the items seems even harder to approximate, as we can show the following by a reduction from IndependetSet. Theorem 4 Consider the revenue maximization problem with affine price functions and limited availability of items. For any ε > 0, it is NP-hard to approximate the maximum revenue to within a factor n1−ε . This result holds even if all bidders have unit valuations, the availability of each item is one, and each item is requested by at most two bidders. References [1] G. Aggarwal, T. Feder, R. Motwani, and A. Zhu, Algorithms for multi-product pricing, Automata, Languages and Programming - ICALP 2004 (J. Dı́az, J. Karhumäki, A. Lepistö, and D. Sannella, eds.), Lecture Notes in Computer Science, vol. 3142, Springer, 2004, pp. 72–83. [2] M.F. Balcan and A. Blum, Approximation algorithms and online mechanisms for item pricing, Proc. of the 7th ACM Conference on Electronic Commerce, ACM, 2006, pp. 29–35. [3] P. Briest and P. Krysta, Single-minded unlimited supply pricing on sparse instances, Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM-SIAM, 2006, pp. 1093–1102. [4] E. D. Demaine, U. Feige, M.T. Hajiaghayi, and M. R. Salavatipour, Combination can be hard: Approximability of the unique coverage problem, Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM-SIAM, 2006, pp. 162–171. [5] V. Guruswami, J. D. Hartline, A. R. Karlin, D. Kempe, C. Kenyon, and F. McSherry, On profit-maximizing envy-free pricing, Proc. 16th Annual ACMSIAM Symposium on Discrete Algorithms, ACM-SIAM, 2005, pp. 1164–1173. [6] J. D. Hartline and V. Koltun, Near-optimal pricing in near-linear time, Algorithms and Data Structures - WADS 2005 (F. K. H. A. Dehne, A. LópezOrtiz, and J.-R. Sack, eds.), Lecture Notes in Computer Sciences, vol. 3608, Springer, 2005, pp. 422–431. [7] D. Lehman, L. I. O’Callaghan, and Y. Shoham, Truth revelation in approximately efficient combinatorial auctions, Journal of the ACM 49 (2002), no. 5, 1–26. [8] P. Rusmevichientong, B. Van Roy, and P. W. Glynn, A nonparametric approach to multi-product pricing, Operations Research 54 (2006), no. 1, 82–98. 60 Operations Research for hospital process optimization Erwin W. Hans Department of Operational Methods for Production and Logistics School of Management and Governance University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands Abstract With a background in manufacturing planning and scheduling, since four years we have ventured in the world of process optimization in hospitals. Due to an ageing population, long waiting lists and increasing costs for healthcare, this research field has received a lot of attention in the last few years. Up to now, the primary focus of this research has been the planning and scheduling of the operating room department, and subsequent departments like Intensive Care and wards. In this research we have intensively collaborated with Erasmus Medical Center, Rotterdam, and Amsterdam Medical Center. In this presentation we present the developed models and algorithms we have developed. Also we discuss our directions of future research, and some ’open problems’ where operations researchers are needed. This research is a collaboration of the departments: Operational Methods for Production and Logistics, Discrete Mathematics and Mathematical Programming, and Stochastic Operations Research. Email address: e.w.hans@utwente.nl (Erwin W. Hans). 61 62 On Path Partitions and Colourings in Digraphs Irith Ben-Arroyo Hartman a a Caesarea Rothschild Institute for Interdisciplinary Applications of Computer Science University of Haifa, Haifa 31905, Israel Key words: path partitions, t-path systems, colourings, t-colourings , Greene’s Theorem, Greene-Kleitman’s Theorem 1 Introduction Dilworth’s well known theorem [6] states that in a partially ordered set the size of a maximum antichain equals the size of a minimum chain partition. Greene and Kleitman [11] generalized Dilworth’s theorem to a min-max theorem in the case when a set of k antichains is considered, for a given integer k ≥ 1. Previously, Greene [10] had proved a similar min-max theorem where the role of chains replaces the role of antichains, and vice versa. Linial [13] conjectured that the theorems of Greene-Kleitman and Greene can be extended to all digraphs by replacing the equality by an inequality. Later, Berge [2] made a stronger conjecture than Linial’s extending the Greene-Kleitman theorem to all digraphs, and Aharoni, Hartman and Hoffman [1] made a similar conjecture which extends Greene’s theorem to all digraphs, and is stronger than Linial’s conjecture. Both conjectures of Berge and Aharoni-HartmanHoffman were proved for all acyclic digraphs (see [13], [5],[15], [4] and [1]). For k = 1 Berge’s conjecture holds by the Gallai-Milgram theorem [9], and the Aharoni-Hartman-Hoffman conjecture holds by the Gallai-Roy theorem [8,14]. Recently, Berger and Hartman [3] proved Berge’s conjecture for k = 2. For other values of k (except for the extreme upper values), all of the conjectures mentioned above are open. For a survey of the subject see [12]. In this talk we give an outline of the proof of Berge’s conjecture for k = 2. We Email address: irith@cs.haifa.ac.il (Irith Ben-Arroyo Hartman). 63 also suggest algorithmic proof techniques of the Aharoni-Hartman-Hoffman conjecture for acyclic digraphs which may be extended for all digraphs. 2 Preliminaries and the Main Result Let G = (V, E) be a directed graph and let |V | = n. If L is a collection of  subsets of V , we set L := {x; x ∈ A for some A ∈ L}. The cardinality of the set X is denoted by |X|. A path P in G is a sequence of distinct vertices (v1 , v2 , . . . , vl ) such that (vi , vi+1 ) ∈ E, for i = 1, 2, . . . , l − 1. Let |P | = l. For positive integers t, k, a t-path system is a family P t := {P1 , P2 , . . . , Pt } of t pairwise disjoint paths, a k-colouring is a family C k := {C1 , C2 , . . . , Ck } of k pairwise disjoint independent sets, also called colour classes.   Denote λt :=max| P t | and αk :=max| C k | where the maximum is taken over all t-path systems and k-colourings, respectively. A t-path system with  | P t | = λt is called optimal. A family P of paths is called a path partition of G if all its members are pairwise disjoint, and ∪P = V . The k-norm |P|k of a path partition P = {P1 , . . . , Pm }  is defined by |P|k := m i=1 min{|Pi |, k}. A partition which minimizes |P|k is called k-optimal. Let πk (G) = min|P|k where the minimum is taken over all possible path partitions in G. Similarly, a colouring C is a family of pairwise disjoint independent sets where ∪C = V . The t-norm |C|t of a colouring C = {C1 , . . . , Cm } is defined by  |C|t := m i=1 min{|Ci |, t}. Let χt (G) = min|C|t where the minimum is taken over all possible colourings in G. If G is a graph of a Poset, then the Greene-Kleitman theorem [11] states that αk (G) = πk (G) for all 1 ≤ k ≤ λ1 and Greene’s theorem [10] states that λt (G) = χt (G) for all 1 ≤ t ≤ α1 . For any graph G, a k-colouring C k is orthogonal to a path partition if each Pi ∈ P meets exactly min{|Pi |, k} different colour classes in C k . Similarly, a colouring C is orthogonal to a t-path system P t if each Ci ∈ C meets exactly min{|Ci |, t} different paths in P t . Conjecture 2.1 (Berge[2]) Let G be a digraph, k a positive integer, and P a k-optimal path partition. Then there exists a k-colouring C k orthogonal to P. Conjecture 2.2 (Aharoni, Hartman, Hoffman [1]) Let G be a digraph, t a positive integer, and P t an optimal t-path system. Then there exists a 64 colouring C orthogonal to P t . Conjecture 2.3 (Linial[13]) Let G be a digraph, and k, t positive integers. Then (1) αk (G) ≥ πk (G) (2) λt (G) ≥ χt (G) Conjecture 2.1 implies part (1) of Conjecture 2.3 and it holds for k = 1 by the Gallai-Milgram [9] theorem. Conjecture 2.2 implies part (2) of Conjecture 2.3 and it holds for t = 1 by the Gallai-Roy [8,14] theorem. The following definition of Frank [7] helps us in uniting both conjectures: Definition 2.4 A t-path system P t = {P1 , P2 , . . . , Pt } and a k-colouring C k = {C1 , C2 , . . . , Ck } are orthogonal if  (1) V = (∪P t ) (∪C k ) (2) |Pi ∩ Cj | = 1 for 1 ≤ i ≤ t, 1 ≤ j ≤ k Observation 2.5 Let P t be a t-path system orthogonal to C k , a k-colouring, for some integers t and k. Then C k is orthogonal to the path partition P := P t ∪ {{x}; x ∈ / ∪P t } and the colouring C := C k ∪ {{x}; x ∈ / ∪C k } is orthogonal t t k to P . Furthermore, if P and C are orthogonal then αk (G) ≥ πk (G) and λt (G) ≥ χt (G), implying Linial’s conjectures for these values of k and t. Theorem 2.6 Conjecture 2.1 holds for all digraphs for k = 2. Proof Outline: The proof is algorithmic. Given a path partition P, we either find a 2−colouring C 2 orthogonal to P, or we find another path partition P ′ with |P ′ |2 < |P|2 . In the proof we use alternating trails relative to P that allow us to define P ′ , or find the 2-colouring C 2 . The proof extends a known proof for the GallaiMilgram Theorem, though it is more complex technically. References [1] R. Aharoni, I. Ben-Arroyo Hartman and A. J. Hoffman, Path partitions and packs of acyclic digraphs, Pacific Journal of Mathematics (2)118 (1985), 249259. [2] C. Berge, k-optimal partitions of a directed graph, Europ. J. Combinatorics (1982) 3, 97-101. 65 [3] E. Berger and I.B.-A. Hartman, Proof of Berge’s Strong Path Partition Conjecture for k = 2, Europ. J. Combinatorics (2007). [4] K. Cameron, On k-optimum dipath partitions and partial k-colourings of acyclic digraphs, Europ. J. Combinatorics 7 (1986), 115-118. [5] K. Cameron, Gallai-type min-max inequalities, Proceedings of the Third ChinaUSA International Conference on Graph Theory, Combinatorics, Computing and Applications, World Scientific, Singapore (1994), 7-16. [6] P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (1), vol.51(1950), 161-166. [7] A. Frank, On chain and antichain families of a partially ordered set, J. Comb. Theory, Ser B, 29 (1980), 176-184. [8] T. Gallai, On directed paths and circuits, in: Theory of Graphs,( P. Erdos and G. Katona, eds.), Academic Press, New York, 1968, 115-118. [9] T. Gallai and A.N. Milgram, Verallgemeinerung eines graphentheoretischen satzes von Redei, Acta Sc. Math. 21 (1960), 181-186. [10] C. Greene, Some partitions associated with a partially ordered set, J. Combinatorial Theory, Ser. A 20 (1976),69- 79. [11] C. Greene, D.J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory, Ser. A 20 (1976), 41-68. [12] I.B.-A. Hartman, Berge’s conjecture on directed path partitions - a survey, Discrete Math. 306 (2006), 2498-2514. [13] N. Linial, Extending the Greene-Kleitman theorem to directed graphs, J. Combinatorial Theory, Ser.A 30 (1981), 331-334. [14] B. Roy, Nombre chromatique et plus longs chemins, Rev. F1, Automat. Informat. 1 (1976), 127-132. [15] M. Saks, A short proof of the k-saturated partitions, Adv. In Math. 33 (1979), 207-211. 66 On a Network Pricing Problem with Consecutive Toll Arcs G. Heilporn a M. Labbé a P. Marcotte b G. Savard c a Université Libre de Bruxelles, Belgique. b Université c École de Montréal, Canada. Polytechnique de Montréal, Canada. Key words: Price setting, mixed integer programming, combinatorial optimization. Consider the tarification problem of maximizing the revenue generated by tolls set on a subset of arcs of a transportation network, where origin-destination flows are assigned to shortest paths with respect to the sum of tolls and initial costs. The Network Pricing Problem with Consecutive Toll Arcs deals with structured networks in which all toll arcs must be connected. As those special structures can represent features specific to a real highway topology, we define a highway as the set of all consecutive toll arcs in a network. A user, which travels from an origin to a destination in this network, can either take the shortest toll free path from its origin to its destination, or take a shortest toll free path from its origin to a given entry point of the highway, stay on the highway from that point to an exit point, and then leave the highway to take a shortest toll free path from that point to its destination. In this latter case, the path he/she chooses is uniquely determined by its entry and exit nodes on the highway. In our setting, users are not allowed to reenter the highway. This assumption will be satisfied in most real highway systems. Linear mixed integer models for this problem are presented in litterature. It has also been proved that the Network Pricing Problem with Consecutive Toll Arcs with a single origin-destination pair or a single toll arc is polynomially solvable. We show that the Network Pricing Problem with Consecutive Toll Arcs is NPhard, using a reduction from 3-SAT. Further, we consider the Network Pricing Problem with Consecutive Toll Arcs involving a complete toll subgraph. By introducing triangle and non decrease inequalities constraints, we propose a very realistic model for the problem, which allows for scale economies. We show that this Constrained Network Pricing Problem with Consecutive Toll Arcs is NP-hard, using a reduction from 3-SAT. We also present some new valid inequalities for the Constrained Network Pricing Problem with Consecutive Toll Arcs, and we propose a corresponding separation procedure. Then 67 we restrict our attention to single commodity problems. Given the new family of valid inequalities, a complete description of the convex hull of solutions for the General Network Pricing Problem with Consecutive Toll Arcs can be pointed out. Further, for the Constrained Network Pricing Problem with Consecutive Toll Arcs, several facets of the corresponding convex hull of solutions are highlighted. Finally, we point out the links between the Network Pricing Problem with Consecutive Toll Arcs and a more standard pricing problem in economics. References [1] S. Dewez, On The Toll Setting Problem, Université Libre de Bruxelles, 2004. [2] M. R. Garey and D. S. Johnson, Computers and intractability : a guide to the theory of NP-completeness, 1979. [3] V. Guruswami, J.D. Hartliney, A.R. Karlin, D. Kempez, C.K. Kenyon and F. McSherry, On Profit-Maximizing Envy-free Pricing, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), Vancouver, Canada, pages 1164-1173, 2005. [4] M. Labbé, P. Marcotte and G. Savard, A Bilevel Model of Taxation and its Application to Optimal Highway Pricing, Management Science, volume 44, pages 1608 - 1622, 1998. 68 On Revenue Equivalence in Truthful Mechanisms Birgit Heydenreich a, Rudolf Müller a, Marc Uetz a, Rakesh Vohra b, a Maastricht University, Quantitative Economics, P.O.Box 616, NL–6200 MD Maastricht, The Netherlands b Northwestern University, Department of Managerial Economics and Decision Sciences, Kellogg Graduate School of Management, Evanston IL 60208 Key words: mechanism design, revenue equivalence, directed graphs, node potential Introduction. In distributed systems, where problem solutions have to be jointly derived by several selfish agents and where problem data is spread over the agents as private information, mechanism design is used to motivate agents to reveal their private information truthfully and to obtain a good overall solution for the system. As a simple example, consider single item auctions, where several bidders are asked to reveal their valuation for a certain good. Dependent on the bids, the mechanism allocates the good to one of the bidders and the price of the good is designed such that agents have an incentive to bid their true valuation. We consider direct revelation mechanisms, which consist of an allocation rule that selects an allocation depending on the agents’ reports about their private information, and a payment scheme that assigns a payment to every agent. Allocation rules that give rise to a mechanism in which truth-telling is a dominant strategy for every agent are called truthfully implementable. Our concern is with the payment scheme that extends a truthfully implementable allocation rule to a truthful mechanism. The property of an allocation rule to have a unique payment scheme completing the allocation rule to a truthful mechanism is called revenue equivalence. We give a characterization for an allocation rule to satisfy revenue equivalence. In order to obtain this characterization, we prove a property on complete directed graphs and apply it to the so called allocation graph, which is defined by the allocation rule and the valuation function of an agent. The characterization holds 1 The first author is supported by NWO grant 2004/03545/MaGW ‘Local Decisions in Decentralised Planning Environments’. 69 for any (possibly infinite) outcome space. Furthermore, we give an elementary and simple proof for the uniqueness of the payment scheme in a truthful mechanism for the case of a finite outcome space under very weak assumptions. The uniqueness of payments in different models is also discussed by e.g. Myerson [7], Green and Laffont [2], Holmström [4], Krishna and Maenner [5], Milgrom and Segal [6], Suijs [10] and Chung and Olszewski [1]. Setting and Basic Concepts. Let the set of agents be denoted by {1, . . . , n} and let A be the (possibly infinite) set of alternative allocations or outcomes. By ti , we denote the type of agent i ∈ {1, . . . , n}, which is an element of the type space Ti ⊆ Rki for some ki ∈ N. Agent i’s preferences over outcomes are modeled by the valuation function vi : A × Ti → R, where vi (a, ti ) is the valuation of agent i for outcome a when he has type ti . A mechanism (f, π) consists of an allocation rule f : ×ni=1 Ti → A and a payment scheme π : ×ni=1 Ti → Rn . In a direct revelation mechanism, the allocation rule chooses for a vector τ of aggregate type reports of all agents an outcome f (τ ), whereas the payment scheme assigns a payment πi (τ ) to each agent i. Let the vector (t−i , ti ) denote the aggregate type report vector when i reports ti and the other agents’ reports are represented by t−i . We assume quasi-linear utilities, that is, the utility of agent i when the aggregate report vector is (t−i , ti ) is vi (f (t−i , ti ), ti )−πi (t−i , ti ). In a truthful mechanism, truth-telling is a (weakly) dominant strategy for every agent: Definition 1 (dominant strategy incentive compatible, truthful) A direct revelation mechanism (f, π) is called dominant strategy incentive compatible or truthful if for every agent i, every type ti ∈ Ti , all aggregate type vectors t−i that the other agents could report and every type si ∈ Ti that i could report instead of ti : vi (f (t−i , ti ), ti ) − πi (t−i , ti ) ≥ vi (f (t−i , si ), ti ) − πi (t−i , si). If for allocation rule f there exists a payment scheme π such that (f, π) is a truthful mechanism, then f is called truthfully implementable. Definition 2 (Revenue Equivalence) A truthfully implementable allocation rule f satisfies the revenue equivalence property if for any two incentive compatible mechanisms (f, π) and (f, π ′ ) and any agent i there exists a function hi that only depends on the reported types of the other agents t−i such that ∀ti ∈ Ti : πi (ti , t−i ) = πi′ (ti , t−i ) + hi (t−i ). Unique Node Potentials in Directed Graphs. Let G = (V, E) be a complete directed graph with (possibly infinite) node set V and arc set E. We assume that G does not contain a negative cycle. By ℓab we denote the (finite) length of the arc from node a to node b. Let a, b ∈ V be two nodes and let P a (finite) path from a to b – or short (a, b)-path – in G. Denote its length by length(P ). For a = b, we regard the path without any edges as (a, b)-path as well and define its length as 0. Define P(a, b) as the set of all (a, b)-paths. Let distG (a, b) = inf P ∈P(a,b) length(P ). If V is a finite set, then distG (a, b) simply 70 equals the length of a shortest path from a to b in G. For infinite V , such a shortest path may not exist. Nevertheless, distG (a, b) is finite, since we assume that G does not have any negative cycle and hence length(P ) + ℓba ≥ 0 holds for every (a, b)-path P . A node potential p is a function p : V → R such that for all arcs (i, j) ∈ E p(j) ≤ p(i) + ℓij . It is well known that the existence of a node potential in a graph is equivalent to the non-existence of negative cycles in that graph. We prove the following characterization for its uniqueness. Theorem 3 Let G = (V, E) be a complete directed graph that does not contain a negative cycle. Then the following statements are equivalent. (1) Any two node potentials in G differ only by a constant. (2) For all a, b ∈ V , distG (a, b) + distG (b, a) = 0. Next, we define a property of the graph G that is sufficient (though not necessary) for uniqueness of node potentials up to a constant. Definition 4 (Two-Cycle Connected) A graph with node set V and arc lengths ℓ is called two-cycle connected if for every partition V1 ∪ V2 = V , V1 ∩ V2 = ∅, V1 , V2 = ∅, there are a1 ∈ V1 and a2 ∈ V2 with ℓa1 a2 + ℓa2 a1 = 0. Theorem 5 Let G be a directed graph without negative cycle. If G is two-cycle connected then its node potential is uniquely defined up to a constant. Characterization of Revenue Equivalence. Fix agent i and let the reports of the other agents be fixed as well. For simplicity of notation we write T and v instead of Ti and vi . We regard f and π as functions of i’s type alone, i.e. f : T → A and π : T → R. Let f be truthfully implementable. Revenue equivalence asserts that any two payment schemes assigning a payment to each type of agent i differ by a constant. If (f, π) is truthful, it is easy to see that for any pair of types s, t ∈ T such that f (t) = f (s) = a for some a ∈ A, the payments must be equal, i.e. π(t) = π(s) = : πa . A payment scheme for agent i is therefore completely defined if the numbers πa are defined for all outcomes a ∈ A such that f −1 (a) is nonempty. Therefore, we may without loss of generality assume that f is onto. As in Gui et al. [3] and Saks and Yu [9], let us define the complete directed and possibly infinite allocation graph Gf with node set A. Between any two nodes a, b ∈ A, there is a directed arc with length ℓab = inf t∈f −1 (b) (v(b, t) − v(a, t)). From Gui et al. [3] and Rochet [8] follows that f is truthfully implementable if and only if Gf does not have a finite cycle of negative length. Let us call this property the nonnegative cycle property. We observe that a payment scheme that complements an allocation rule f to a truthful mechanism can be interpreted as a node potential in Gf . Hence, a truthfully implementable allocation rule f satisfies revenue equivalence if and only if in Gf the node potential is uniquely defined up to a constant. Note that the existence of a node potential is already guaranteed by the nonnegative cycle property. The characterization 71 of its uniqueness up to a constant follows immediately from Theorem 3. Theorem 6 Let f be a truthfully implementable allocation rule. Then f satisfies revenue equivalence iff distGf (a, b) + distGf (b, a) = 0 for all a, b ∈ A. For finite outcome spaces, we prove that revenue equivalence is satisfied under very weak conditions, which cannot be relaxed. More specifically, the following result follows from Theorem 5. Theorem 7 Let A be a finite outcome space. Let each agent i ∈ {1, . . . , n} have types from the (topologically) connected type space Ti ⊆ Rki . Let each agent’s valuation function vi (a, ·) be a continuous function in the type of the agent for every a ∈ A. Then, every truthfully implementable allocation rule f satisfies revenue equivalence. References [1] K-S. Chung and W. Olszewski. A non-differentiable approach to revenue equivalence. Working Paper, Northwestern University, Evanston, IL, 2006. [2] J. Green and J.-J. Laffont. Characterization of satisfactory mechanisms for the revelation of preferences for public goods. Econometrica, 45(2):427–438, 1977. [3] H. Gui, R. Müller, and R. Vohra. Dominant strategy mechanisms with multidimensional types. Discussion Paper 1392, The Center for Mathematical Studies in Economics & Management Sciences, Northwestern University, Evanston, IL, October 2004. [4] B. Holmström. Groves’ scheme on restricted domains. 47(5):1137–1144, 1979. Econometrica, [5] V. Krishna and E. Maenner. Convex potentials with an application to mechanism design. Econometrica, 69(4):1113–1119, 2001. [6] P. Milgrom and I. Segal. Envelope theorems for arbitrary choice sets. Econometrica, 70(2):583–601, 2002. [7] R. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58–73, 1981. [8] J.-C. Rochet. A necessary and sufficient condition for rationalizability in a quasi-linear context. Journal of Mathematical Economics, 16(2):191–200, 1987. [9] M. Saks and L. Yu. Weak monotonicity suffices for truthfulness on convex domains. In Proc. 6th ACM conference on Electronic Commerce, pages 286 – 293, 2005. [10] J. Suijs. On incentive compatibility and budget balancedness in public decision making. Economic Design, 2:193–209, 1996. 72 p A Multi-phase Approach to the University Course Timetabling Problem ⋆ Shahadat Hossain ∗ Minhaz F. Zibran Department of Mathematics and Computer Science University of Lethbridge, Canada. Key words: Constraint Programming, Course Scheduling, Graphical User Interface. 1 Introduction Course timetabling is a well-known combinatorial problem that is needed to be solved regularly at educational institutions. There are a number of general constraints applicable to course timetabling problem in any organization. Depending on the institution, additional constraints and preferences may also apply. This paper presents a multi-phase implementation of the timetabling problem using constraint programming technology. The software implementation incorporates a commercial solver ILOG’s CPLEX, Java technology for graphical user interface, and C++ for piecing together of software components. We solve the problem in a sequence of four phases. Phase-1 assigns instructors to courses (lectures, labs and tutorials), phase-2 assigns lectures to days, phase3 assigns lectures to time-slots and finally phase-4 assigns labs and tutorials to days and available time-slots in the days. Each phase is solved using constraint programming with suitable heuristics for ordering the decision variables and maximizes an objective function over a given set of constraints and preferences. A useful feature of our implementation is that it allows the user to customize constraints as well as to generate new solutions that may incorporate partial solutions from perviously generated feasible solutions. ⋆ This research was supported in part by NSERC discovery grant. ∗ Corresponding Author. Email addresses: shahadat.hossain@uleth.ca (Shahadat Hossain), minhaz.zibran@uleth.ca (Minhaz F. Zibran). 73 2 The Problem and Solution Approach Timetabling problems have been traditionally solved using greedy heuristics, meta-heuristics such as hill climbing and tabu search, integer linear programming[1], and more recently using constraint programming techniques[2]. Cambazard et al.[3] proposed an algorithm for solving course timetabling problem using constraint programming that allows automatic relaxations of constraints. Lotfi, Vahid and Cerveny[4] introduced a multi-phase algorithm to solve a final exam scheduling problem. In our work we propose a multi-phase algorithm to solve the course timetabling problem using constraint programming. In what follows we use the following terminologies to describe the problem and solution strategies. Topic: A topic is a set of related lectures, labs and tutorials. For example, “CS1000: Computer Basics” is a topic, which may include two lectures, three labs and two tutorials. Course: A course refers to a single lecture or lab or tutorial. Instructor: An instructor is a person who teaches/conducts a course. There may be different types of instructors, such as Professors, Academic Assistants and Graduate Assistants (Teaching Assistants). Week-day: A week-day is a working day of a week from Monday through Friday. Day-Sequence: A day-sequence is a set of week-days. Week-days are divided into two day-sequences: MWF (Monday, Wednesday and Friday) and TR (Tuesday and Thursday). Time-slot: A time-slot refers to the unit of time-span specified by starting and ending time. Each week-day is divided into a fixed number of time-slots. Each time-slot may be allotted to courses to be taught during that period. Duration of 3 time-slots in MWF equals to the duration of 2 time-slots in TR. One year prior to the start of an academic semester, each academic department at the University of Lethbridge determines the courses to be offered in that semester. Instructors mention which courses they would like to teach with a numerical preference level for each course. Instructors also mention the day or day-sequence and time of day (morning or afternoon) they prefer to teach. The department then prepares a timetable assigning courses to instructors, days or day-sequences and time-slots respecting the individual preferences as much as possible. The constraints and preferences specific to our problem include the following. • Professors conduct only lectures, academic assistants conduct labs and tutorials, and graduate students conduct only labs. Each instructor has an upper limit on the number of courses the instructor may teach. 74 Fig. 1. A snapshot of user interface of the timetabling implementation. • A lecture has to be scheduled in one of the two day sequences. When a lecture is assigned to a day-sequence, it is taught at the same time in each day of the day sequence. • A lab or tutorial has to be assigned to only one of the five week-days. • Lectures and associated labs and tutorials of a single topic have to be scheduled in such a way that for each lecture at least one lab and tutorial are available in non-overlapping time-slots. • There should be a gap of at least one time-slot between courses taught by the same instructor. • The preference of instructors on course, day, and time should be satisfied as much as possible. We decompose the course scheduling problem into four phases and each phase is solved separately using constraint programming solvers. In phase-1, courses (lectures, labs, and tutorials) are assigned to the instructors (professors, academic assistants and graduate students). phase-2 assigns lectures to one of the two day-sequences. In phase-3, lectures of a single day-sequence are assigned to the time-slots available on that day-sequence. And finally phase-4 assigns labs and tutorials to week-days and available time-slots within the week-days. At each phase, the objective is to maximize the preferences (given as an objective function) subject to the given constraints. One of the objectives of this work is to allow the users enough flexibility so that customized schedules can be produced in a user-friendly way. The customization allowed in the current implementation includes addition or removal of constraints on the fly, loading and modifying a previously saved solution, and computing a new solution from a partial solution. In such cases, the new solution is attained quickly as it is not necessary to solve the problem from scratch[5]. 75 3 Experiments and Evaluation We experimented with our timetabling package in Windows XP environment on a AMD Athlon(tm) 64 processor 3500+ with 512 MB RAM. Our test data included 120 courses (35 lectures and 85 are labs and tutorials), 34 instructors (22 Professors, 9 Academic Assistants and 3 Graduate Students), 20 classrooms, 8 time-slots in each day of MWF and 6 time-slots in each day of TR. At each run on the data a feasible solution for each phase was obtained in less than two minutes. 4 Concluding Remarks Course timetabling problem is a constraint satisfaction optimization problem. Therefore, constraint programming techniques are well-suited for solving such problems, as it exploits the structure of the problem. Multi-phase approach divides the complexity of the original problem into several subproblems of reduced complexity. Since the phases are solved separately, partial solutions may be generated, examined and amended. New solutions can be computed quickly from the partial solutions. Furthermore, the graphical user interface (see Figure 1) and the provision for dynamically adding or removing constraints make our timetable package flexible and user-friendly. References [1] MirHassani, S.A. A Computational Approach to Enhancing Course Timetabling with Integer Programming. Applied Mathematics and Computation, 179(2): 814 – 822, 2006. [2] Hentenryck, Pascal Van and Saraswat, Vijay. Constraint Programming: Strategic Directions. Constraints: An International Journal, Vol. 2, No. 1, pp. 7 – 33 (1997). Kluwer Academic Publishers. [3] Cambazard, Hadrein and et al. Interactively Solving School Timetabling Problems Using Extensions of Constraint Programming. E. Burke and M. Trick (Eds.): PATAT 2004, LNCS 3616, pp. 190 – 207, 2005. [4] Lotfi, Vahid and Cerveny, Robert. A final Exam Scheduling Package. J. Opl. Res. Soc. Vol. 42, No. 3, pp. 205 – 216, 1991. [5] T. Müller, R. Bartàk, H. Rudovà. Minimal Perturbation Problem in Course Timetabling. Practice and Theory of Automated Timetabling, Selected Revised Papers, pp. 126 – 146. Springer-Verlag LNCS 3616, 2005. 76 Approximating Minimum Independent Dominating Sets in Wireless Networks Johann L. Hurink a Tim Nieberg b a University of Twente Faculty of Elec. Engineering, Mathematics & Computer Science P.O.Box 217, NL-7500 AE Enschede b Research Institute for Discrete Mathematics University of Bonn Lennéstr. 2, D-53113 Bonn Key words: Minimum Independent Dominating Set, Minimum Maximal Independent Set, PTAS, Wireless Communication Graph, Bounded Growth 1 Introduction We present and discuss a Polynomial-Time Approximation Scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth that are used to model wireless networks. Two vertices are called independent if they are not adjacent to each other, and a subset I ⊆ V is called independent if all its vertices are not connected. A subset D ⊆ V is called dominating if every vertex from V is contained in this subset, or adjacent to a vertex from D. In the following, we seek for a small subset of vertices that is both independent and dominating. Such a set combines thus the advantages of both structures, and many application in the area of wireless ad hoc networking rely on these structures. The resulting problem is called Minimum Independent Dominating Set (Min-IDS) problem and it is sometimes also called Minimum Maximal Independent Set problem. A communication network is modeled as an undirected graph G = (V, E), where two vertices are adjacent if they can communicate with one another directly. Consequently, a wireless network is created by the direct communication links between a collection of wireless transceivers. The nature of wireless Email addresses: J.L.Hurink@utwente.nl (Johann L. Hurink), Nieberg@or.uni-bonn.de (Tim Nieberg). 77 Algorithm 1 Dominating Set. Input: G = (V, E) poly. growth-bounded, ε > 0 Output: Dominating Set D 1: D := ∅; i := 0; 2: while V = ∅ do 3: Pick v ∈ V ; 4: ri := 0; (i) 5: while |Dri+3 (v)| > (1 + ε) · |Dr(i)i (v)| do 6: ri := ri + 1; 7: end while (i) 8: Color vertices in Dri +3 (v) with color i; (i) D := D ∪ Dri +3 (v); 9: 10: V := V \ Γri +3 (v); 11: i := i + 1; 12: end while transmissions however leads to more structured graph, which we capture by the definition of polynomially bounded growth. Definition 1 Let G = (V, E) be a graph. If there exists a polynomial p(.) of bounded maximal degree such that every r-neighborhood in G contains at most p(r) independent vertices, then G is polynomially p-growth bounded. Here, we denote by Γr (v) := {u ∈ V | dG (u, v) ≤ r} the r-neighborhood of a vertex v ∈ V , and we omit the center vertex v in case it is unambiguous. The above definition includes many well-known classes of graph models used for wireless networks, e.g. (Quasi) Unit Disk Graphs or Coverage Area Graphs [1,3]. Note that the definition does not rely on any geometric information. For an r-neighborhood, we further define Dr := D(Γr ) to be a Min-IDS for all vertices in Γr (computed with respect to G), and we denote an optimal solution for the graph G by D ∗ . An algorithm that, for any ε > 0, runs in polynomial time, and that always returns a feasible solution of relative error no more than 1 + ε is called Polynomial-Time Approximation Scheme (PTAS). In our case, we present a PTAS for the Min-IDS problem that returns an independent and dominating set of cardinality at most 1 + ε times the cardinality of a minimum independent dominating set. Note that the runtime of such an algorithm is allowed to depend on the parameter ε, but should be polynomial for fixed ε > 0. Our approach works in two stages. First, we locally exploit the properties of bounded growth graphs to create partial, local solutions, and then combine these to a globally feasible solution. 78 2 Local Solutions Consider Algorithm 1, the main part iteratively constructs independent dominating sets for growing neighborhoods Γr , and stops increasing the radius r if |Dr+3 | ≤ (1 + ε) · |Dr | holds. Then, the solution D and the set of remaining vertices is updated. Based on earlier work given in [4], we can compute the local solution sets 1 1 Dr in time nO( ε log ε ) , especially since the largest radius r to be considered is bounded by a respective constant that solely depends on ε. It is easy to see that the set D dominates G, and further we have the following lemma. |D| ≤ (1 + ε)|D ∗|. Lemma 2 [4] 3 Global Feasibility While the set D dominates the graph G and satisfies the desired approximation guarantee, there may still be edges present so that D is not an independent set. If that is the case, we resort to Algorithm 2 which removes conflicting, i.e. non-independent, vertices and adds independent ones in order to create an independent set. Lemma 3 Let G be of polynomially bounded growth. A single invocation of Algorithm 2 removes all conflicts involving the argument vertex v, and increases the cardinality of D by at most p(1). To repair a solution, we create a candidate set C ⊂ D by adding conflicting vertices according to the order given by their colors. We then have the following observation. Lemma 4 |C| ≤ ε|D ∗|. After having repaired all conflicts in C the overall independent dominating set D constructed by the approach then is of cardinality |D| ≤ |D| + (p(1) − 1)|C| ≤ (1 + p(1) · ε)|D ∗|, which gives the PTAS. The runtime of the algorithm is dominated by the 1 1 computation of the local subsets Dr , and is thus nO( ε log ε ) . 79 Algorithm 2 Repair Independence (v). Input: G = (V, E), Dominating Set D, v ∈ D 1: D := D \ {v}; 2: V ′ := V \ Γ(D); 3: Compute independent dominating set I on G[V ′ ]; 4: D := D ∪ I; 4 Conclusions The presented PTAS can be modified to yield a robust algorithm [5]. Generally speaking, a robust algorithm accepts any type of input, i.e. not necessarily a graph of polynomially bounded growth, and always computes meaningful output. In our case, the approximation scheme then accepts any graph as input, and either returns a Minimum Independent Dominating Set which satisfies the (1 + ε) approximation bound, or a certificate that shows that the input instance does not fulfill the property of polynomially bounded growth. Note that in case the input is a graph of polynomially bounded growth, the algorithm always returns a valid solution. Since Unit Disk Graphs are also of polynomially bounded growth, our result improves the currently known 5-approximation for the Min-IDS problem on this graph class [2], and is also applicable to a larger class of graphs. Altogether, we conclude with the following theorem: Theorem 5 There exists a PTAS for the Min-IDS problem on polynomially bounded growth graphs. The PTAS only requires adjacency information of the input graph, and is robust in respect to other input instances. References [1] B.N. Clark, C.J. Colburn, and D.S. Johnson. Unit Disk Graphs. Discrete Mathematics, 86:165–177, 1990. [2] P. Crescenzi and V. Kann (Eds.). A compendium of NP optimization problems. http://www.nada.kth.se/ viggo/problemlist/. [3] T. Nieberg and J.L. Hurink. Wireless Communication Graphs. Proc. Intelligent Sensors, Sensor Networks and Information Processing Conf. 367–372, 2004. [4] T. Nieberg, J.L. Hurink, and W. Kern. Approximation Schemes for Wireless Networks. Preprint, 2006. [5] V. Raghavan and J. Spinrad. Robust algorithms for restricted domains. Journal of Algorithms, 48(1): 160–172, 2003. 80 Representation of Poly-antimatroids Yulia Kempner ∗ Department of Computer Science Holon Institute of Technology Holon, Israel Vadim E. Levit Department of Computer Science and Mathematics The College of Judea and Samaria Ariel, Israel Key words: Abstract convexity, multiset, poly-antimatroid. 1 Preliminaries An antimatroid is an accessible set system closed under union [2]. An algorithmic characterization of antimatroids based on the language definition was introduced in [3]. Later, another algorithmic characterization of antimatroids which depicted them as set systems was developed in [4]. While classical examples of antimatroids connect them with posets, chordal graphs, convex geometries, etc., a game theory gives a framework, in which antimatroids are considered as permission structures for coalitions [1]. A poly-antimatroid is a generalization of the notion of the antimatriod for multisets [5]. In this research we concentrate on interrelations between geometric, algorithmic, and lattice properties of poly-antimatroids that leads us to interesting results in abstract convexity. Let E be a finite set. A multiset A over E is a function fA : E → N, where fA (e) is a number of repetitions of an element e in A. A multiset system over E is a pair (E, F ), where F is a family of multisets over E, called feasible multisets. ∗ Corresponding author Email addresses: yuliak@hit.ac.il (Yulia Kempner), levitv@yosh.ac.il (Vadim E. Levit). 81 Definition 1.1 A finite multiset system (E, F ) is a poly-antimatroid if (A1) F is an accessible system, i.e., for each non-empty X ∈ F , there is x ∈ X such that X − x ∈ F (A2) F is closed under union (X, Y ∈ F ⇒ X ∪ Y ∈ F ) 2 Main Results We consider a geometric characterization of two-dimensional poly-antimatroids. Let E = {x, y}. In this case, a poly-antimatroid may be represented as a set of points on the digital plane N2 , since each A ∈ F is defined by the pair (fA (x), fA (y)), which may be denoted as a point (xA , yA ) in N2 . If A = (x, y) is a point in a digital plane, the 4-neighborhood N4 (x, y) is the set of points N4 (x, y) = {(x − 1, y), (x, y − 1), (x + 1, y), (x, y + 1)}, and the 8-neighborhood N8 (x, y) is the set of points N8 (x, y) = {(x − 1, y), (x, y − 1), (x + 1, y), (x, y + 1), (x − 1, y − 1), (x − 1, y + 1), (x + 1, y − 1), (x + 1, y + 1)}. Let m be either 4 or 8. A sequence A0 , A1 , ..., An is called a Nm -path if Ai ∈ Nm (Ai−1 ) for each i = 1, 2, ..., n. Any two points A, B ∈ S are said to be Nm connected in S if there exists a Nm -path A = A0 , A1 , ..., An = B from A to B such that Ai ∈ S for each i = 0, 1, ..., n. A digital set S is a Nm -connected set if any two points P ,Q from S are Nm -connected in S. A Nm -connected component of a set S is a maximal Nm -connected subset of S. Proposition 2.1 A poly-antimatroid (E, F ) is N4 -connected component in the digital plane N2 . Moreover, N2 − F is N4 -connected as well. Hence, each poly-antimatroid can be represented by its boundary. A point A in the set S is called interior if N8 (A) ∈ S. A point in S which is not an interior point is called a boundary point. All boundary points of S form the boundary of S. It is possible to find the boundary of any poly-antimatroid using the following algorithm. Suppose we have an oracle that can decide whether each point (x, y) belongs to the poly-antimatroid. Thus, we begin with the point (0, 0), since this point belongs to the boundary of any poly-antimatroid. Then we track the 82 boundary up to the maximum point of the poly-antimatroid Amx = (xmx , ymx ) for which (xmx + 1, ymx ) ∈ / F and (xmx , ymx + 1) ∈ / F . To complete the task we continue to follow the boundary until the point (0, 0). The obtained sequence B may be divided into two parts : the lower part Blower (from the starting point, B0 = (0, 0) to the maximum point Amx ) and the upper part Bupper (from the maximum point Amx to the last point Bn = (0, 0)). y✻ B10 = B26 s s s s s B0 B1 s s s s s s s s s s s s s s s s s s s s s s s s s s B18 s = Amx s s B8 ✲ x B4 Fig. 1. A boundary of a poly-antimatroid. Any two-dimensional poly-antimatroid may be represented by its boundary in the following form: F = Blower ∨ Bupper = {X ∪ Y : X ∈ Blower , Y ∈ Bupper } Since a poly-antimatroid is closed under union, it can be obtained as a union of all its irreducible elements. A multiset A ∈ F is called irreducible if A = X ∪Y , where X, Y ∈ F , implies A = X or A = Y . It is easy to see that each interior point A = (x, y) ∈ F is not irreducible, since from N8 (A) ∈ F it follows that A = (x − 1, y) ∪ (x, y − 1). There exist both irreducible boundary points (see B4 and B8 in Fig. 1) and non-irrreducible boundary points (see B18 in Fig. 1). A particular case of the Krein-Milman theorem states that given a convex polygon, one needs to know only the corners (extreme points) of the polygon to recover its shape. A straightforward analogy can be made between irreducible points and extreme points. Our observation is that the geometrical structure of poly-antimatroids motivates another notion of the extreme point, which is based on the definition of extreme points in convex geometry. In the following we define an extreme point of a poly-antimatroid F as a point X ∈ F such that F − X is a poly-antimatroid as well. For example, in Fig. 1 the boundary points B4 and B8 are extreme, while the boundary points B1 and B10 are not extreme. In the general case, extreme points do not allow us to reconstruct a poly-antimatroid. For instance, the poly-antimatroid in Fig. 1 can not be reconstructed without the point B10 , which is not extreme. Thus, we give a new definition of corner points. Definition 2.2 A corner point of a poly-antimatroid F is a point (x, y) ∈ F such that either (x, y − 1), (x + 1, y) ∈ / F or (x − 1, y), (x, y + 1) ∈ / F. 83 Obviously, the set of corner points includes the set of extreme points. Moreover, each corner point is irreducible as well. Theorem 2.3 Any poly-antimatroid is uniquely determined by its set of corner points. Poly-antimatroid Construction Algorithm. Input: C - a set of points in N2 . Output: if C is the set of corner points of some poly-antimatroid, the algorithm returns the boundary of the poly-antimatroid, otherwise it informs about the negative result. 3 Conclusions We have presented an approach to poly-antimatroid polyhedra which is based both on the notions gathered from computer vision and the insights derived from convex geometry. It is worth noticing that the classical notion of convexity is somewhat limited because it does not allow for differentiation between irreducible and extreme points. When poly-antimatroids are under investigation, the only constraint is the following series of inclusions: Boundary(F ) ⊇ Irreducible(F ) ⊇ Corner(F ) ⊇ Extreme(F ), where Boundary(F ), Irreducible(F ), Corner(F ), and Extreme(F ) are the set of boundary points, the set of irreducible points, the set of corner points, and the set of extreme points of a poly-antimatroid F , respectively. References [1] Bilbao J. M., Cooperative games under augmenting systems, SIAM Journal of Discrete Mathematics 17 (2003), 122–133. [2] Björner A., and G. M. Ziegler, Introduction to greedoids, in Matroid applications, ed. N. White, Cambridge University Press, Cambridge, UK,1992. [3] Boyd E. A., and U. Faigle, An algorithmic characterization of antimatroids, Discrete Applied Mathematics 28 (1990) 197-205. [4] Kempner Y., and V. E. Levit, Correspondence between two antimatroid algorithmic characterizations, The Electronic Journal of Combinatorics 10 (2003). [5] Nakamura M., Characterization of polygreedoids and poly-antimatroids by greedy algorithms, Oper.Res. Lett. 33, no. 4 (2005), 389–394. 84 On full components for Rectilinear Steiner tree Walter Kern 1 Xinhui Wang 2 University of Twente, Department of Applied Mathematics, Faculty of EEMCS, P.O.Box 217, 7500 AE Enschede, The Netherlands Abstract The Rectilinear Steiner tree problem (RST) is to find a shortest tree which connects all the terminal points in a grid graph. The currently fastest algorithms for RST proceed by composing an optimal tree from certain candidate components (tree stars). In this paper, we will present several additional constraints which provably reduce the number of candidate components in the worst case. Key words: Rectilinear Steiner tree problem, full component, Hwang tree, Tree star The Steiner tree problem is a classical NP-hard problem. Given an undirected graph G = (V, E), a weight w(e) > 0 for each edge e ∈ E, and the terminal set Y ⊆ V with |Y | = k, the Steiner tree problem is to find a tree T ∗ with minimum cost which connects all the terminal points in Y . Since the edge weights are positive, the optimal tree T ∗ is a tree with leaves in Y . In the Rectilinear Steiner tree problem (RST), the distance between two points is the sum of differences in x− and y− coordinates. RST gained new importance in the development of techniques for VLSI. The most well known exact algorithm for ST (RST) is the so called Dreyfus-Wagner ([1]) algorithm, a certain dynamic programming approach that computes an optimum tree T ∗ in time O ∗ (3k ). The notion O ∗ suppresses polylogarithmic factors. Recently, the dynamic programming bound was improved to O ∗ (2.684) ([2]), and further in Fuchs et al. ([3]) to O ∗ ((2 + ǫ)k ) with any fixed ǫ > 0. However, these Email addresses: kern@math.utwente.nl (Walter Kern), xinhuiw@math.utwente.nl (Xinhui Wang). 1 Supported by BRICKS. 2 Supported by Netherlands Organization for Scientific Research (NWO) grant 613.000.322 (Exact Algorithms). 85 algorithms are only purely theoretical and are not expected to be helpful in practice. In general, every terminal of an optimal Steiner tree T ∗ is not necessary a leaf. The points in T ∗ which are not terminals are called Steiner points, T is called a full Steiner tree if all the terminals in T are leaves in T . Terminals with degree more than one split T into edge-disjoint full Steiner trees with only the split terminal in common. These trees are called full components. Based on full components, some special algorithms have been proposed for RST. The breakthrough in technique is proposed by Winter ([4]) on Euclidean Steiner problem (EST), and introduced by Salowe and Warme ([5]) to RST. There are two steps in this algorithm. First step is to generate a sufficient number of full components. The second step is to construct the optimal tree using dynamic programming or Branch and Bound. For both approaches, the number of full components in the first step is the main parameter in the running time. We will show this algorithm later. A well known result of Hwang ([6]) then states the existence of an optimum Steiner tree T ∗ = T ∗ (Y ) with each component of the following form (Hwang topology): There are two special terminals, the root r and the tip t of the component, connected to each other by a horizontal and vertical line segment. The line [r, c] is called the long leg or (Steiner ) chain, [t, c] is called the short leg of the component. The chain has an arbitrary number of straight line segments attached to it from both sides alternatingly, each connecting exactly one terminal to the chain. In addition, there may be one exceptional terminal connected to the short leg (cf. Fig.1). r c t Fig. 1. A Hwang tree In what follows, a Steiner tree (component) with Hwang topology as above will be simply called a Hwang tree. A Hwang set is a set X ⊆ Y which is the terminal set of at least one Hwang tree. We let H(X) denote the shortest Hwang tree for X. In the literature, Hwang sets/trees are also known as full sets and full components, as Hwang trees are candidates for T ∗ -components. Ganley and Cohoon ([7]) present a straightforward dynamic program computing an optimum Steiner tree T ∗ by composing T ∗ from Hwang trees in the cheapest way: 86 • Compute H(X1 ) for all X1 ⊆ Y . • Compute recursively for all X ⊆ Y T ∗ (X) := min X=X0 ✶X1 T ∗ (X0 ) ∪ H(X1 ), where X = X0 ✶ X1 ⇔ X = X0 ∪ X1 , and |X0 ∩ X1 | = 1. In [7], it is shown that there are (modulo polynomial factors) at most 1.62k Hwang sets X1 ⊆ Y . More generally, every X ⊆ Y of size i ≤ k has at most 1.62i Hwang subsets X1 ⊆ X. So the above dynamic program has a running time of order   k  k ∗ O ( 1.62i ) = O ∗(2.62k ). i i=1 From the above algorithm, we can see that the number of components which could be part of the optimal tree is crucial for the performance of the dynamic programming. Ganley and Cohoon ([7]) obtain the upper bound of 1.62k for the number of full component. Fößmeier and Kaufmann ([8]) introduced the concept of tree stars (Hwang set with certain additional properties), thereby reducing the number of candidate sets to something between 1.32k and 1.38k . The exact bound of 1.357k has been proven in [9]. Here we present a strengthening of the tree star concept, based on the obvious fact that each candidate component must be an optimal Steiner tree itself. Thus additional condition can be shown to further reduce the number of candidate sets in the worst case (to O ∗ (1.336k )). References [1] S. E. Dreyfus and R. A. Wagner, (1972) The Steiner problem in graphs. Networks, 1, 195–207. [2] B. Fuchs, W. Kern and X. Wang, (2007) Speeding up the Dreyfus-Wagner algorithm for minimum Steiner trees, to appear in Mathematical Methods of Operations Research. [3] B. Fuchs, W. Kern, D. Mölle, S. Richter, P. Rossmanith and X. Wang, (2007) Dynamic programming for minimum Steiner trees, to appear in Theory of Computing Systems. [4] P. Winter, (1985) An algorithm for the Steiner problem in the Euclidean plane. Networks, 15:323-345. [5] J. S. Salowe and D. M. Warme, (1995) Thirty-Five point Rectilinear Steiner minimal trees in a day, Networks, 25, 69-87. 87 [6] F. K. Hwang, (1976) On Steiner Minimal Trees with Rectilinear Distance. SIAM Journal on Applied Mathematics, 30(1), 104–114. [7] J. L. Ganley and J. P. Cohoon, (1994) Optimal rectilinear Steiner minimal trees in O(n2 2.62n ) time, in: Proceedings of the Sixth Canadian Conference on Computational Geometry, 308–313. [8] U. Fößmeier and M. Kaufmann, (2000) On exact solutions for the rectilinear Steiner tree problem Part 1: Theoretical results. Algorithmica, 26, 68–99. [9] B. Fuchs, W. Kern and X. Wang, (2006) The number of tree stars is O∗ (1.357k ), Electronic Notes in Discrete Mathematics, 25, 183-185 . 88 St-serie decomposition of orders Jimmy Leblet 1 Jean-Xavier Rampon 2 Key words: graph, clique cutset, decomposition, orders, partially ordered set. We present a new decomposition, the St-serie decomposition, which generalizes the classical series decomposition of orders. Using a minimality criterion we restrict our study to particular St-serie decompositions thus obtaining some structural properties having interesting algorithmic consequences. If we obtain an algorithm with a linear time complexity, in the size of the input, for deciding if an order admits a St-serie decomposition, we also improve the recognition of the class of bisplit graphs: a class of graph introduced in a different context by A. Brandstädt et al. in a 2005 paper. We just recall the minimal notations and definitions necessary for the reading of this extended abstract, complementary notions about orders can be found, for example, in [4]. First of all, given any strictly positive integer k, we denote the set {1, · · · , k} by [k]. A finite order P is an ordered pair (V (P ), <P ), where V (P ) is a finite set, called the groundset of P , and <P is an irreflexive, transitive binary relation on V (P ). Let x, y ∈ V (P ) and let A, B ⊆ V (P ). When x and y are distinct, they are incomparable, denoted by x P y, if neither x <P y nor y <P x. The set A is an antichain if its elements are pairwise incomparable. The cocomparability graph of P , is the couple G(P ) = (V (P ), {{x, y} : x, y ∈ V (P ) : x = y and x P y}). The predecessors set of x is the set ↓[P x = {y ∈ V (P ) : y <P x}, and the successors set of x is the set ↑[P x = {y ∈ V (P ) : x <P y}. We denote ↓∩P A, respectively ↑∩P A, the set   [ [ x∈A ↑P x. A <P B means that ∀a ∈ A, ∀b ∈ B holds x∈A ↓P x, respectively a <P b. 1 The St-serie decompositions of orders Definition 1 An order P admits (X1 , Z, X2) for St-serie decomposition if (i) its groundset is the disjoint union of these three sets, (ii) X1 and X2 are none 1 Université d’Orléans, L.I.F.O. Bat IIIA, Rue Léonard de Vinci, B.P. 6759, F45067 Orléans Cedex 2, France; jimmy.leblet@univ-orleans.fr 2 FST de l’Université de Nantes, 2, rue de la Houssinière, BP 92208 44322 Nantes Cedex 3, France; jean-xavier.rampon@univ-nantes.fr 89 void, (iii) X1 <P X2 , and (iv) Z is an antichain. If so, P is then said to be St-serie decomposable. Since the classical series decomposition simply corresponds to a St-serie decomposition of type (X1 , ∅, X2 ), it is not surprising to obtain a characterization of St-serie decomposable orders by their cocomparability graphs. Just recall that a clique cutset in graph is a clique whose deletion strictly increases the number of its connected components. Proposition 1 An order P admits a St-serie decomposition if and only if its cocomparability graph is either disconnected or has a clique cutset. For the study of the property ”being St-serie decomposable”, this characterization has for interesting and immediate consequence firstly to insure its comparability invariance, and, secondly to establish the polynomial tractability of its associated decision problem. The latter point can be directly deduced from the result of Whitesides [6] showing that graphs with clique cutsets are polynomially recognizable: see also Tarjan [5] for a O(|V (G)| × |E(G)|) time complexity algorithm, given an input graph G. 2 Minimal St-serie decompositions Using some minimality criterion we obtain an interesting restriction of the St-serie decomposition. Definition 2 Let P be an order having (X1 , Z, X2) for St-serie decomposition. This St-serie decomposition is a minimal St-serie decomposition if it doesn’t exist a St-serie decomposition (X1′ , Z ′ , X2′ ) of P such that Z ′  Z, X1 ⊆ X1′ and X2 ⊆ X2′ . Notice, as easily obtained with a 3 elements chain, that two distinct minimal St-serie decompositions (X1 , Z, X2) and (X1′ , Z ′ , X2′ ) can be such that Z = Z ′ . Here, follows an easy proposition which characterize the minimal St-serie decompositions and which, together with its corollary, appears to be a key point for the sequel. Proposition 2 Let (X1 , Z, X2) a St-serie decomposition of an order P . This decomposition is a minimal one if and only if we have both (↓∩P X2 ) ∩ Z = ∅ and (↑∩P X1 ) ∩ Z = ∅. Corollary 1 Let (X1 , Z, X2 ) a St-serie decomposition of an order P , then there exists (X1′ , Z ′ , X2′ ) a minimal one such that Z ′ ⊆ Z, X1 ⊆ X1′ and X2 ⊆ X2′ . Notice, as illustrated in Figure 1, that several distinct minimal St-serie decompositions can be obtained from a given one. We conclude by some structural 90 X1 X2 X2 (a) X1 X1 X2 (b) (c) Fig. 1. The St-serie decomposition given in (a) has two distinct minimal St-serie decompositions fulfilling Corollary 1. properties. Lemma 2 Let P be a St-serie decomposable order, then for every (X1 , Z, X2 ) minimal St-serie decomposition of P , we have that (i) X2 = {x : x ∈ V (P ) : |↓[P x| ≥ |X1 |} and (ii) X1 = {x : x ∈ V (P ) : |↑[P x| ≥ |X2 |}. Moreover, we have that (i) X2 uniquely determines X1 and Z and (ii) X1 uniquely determines X2 and Z. Corollary 3 The number of minimal St-serie decompositions of an order P is at most |V (P )| − 1 and the bound is tight. Theorem 4 The minimal St-serie decompositions of an order P can be totally ordered by inclusion on their X1 set’s (or equivalently on their X2 set’s). 3 Recognition of St-serie decomposable orders Using the results presented in section 2, we can establish a new characterization, of St-serie decomposable orders, outlining the skeleton of our recognition algorithm. Theorem 5 An order P is St-serie decomposable if and only if there exists l ∈ [|V (P )| − 1] such that for V2 = {x : x ∈ P : |↓[P x| ≥ l}, V1 = ↓∩P V2 and V3 = V (P ) \ (V2 ∪ V1 ) we have that (i) V2 = ∅, (ii) |V1 | = l, and (iii) ∀ x ∈ V3 , ↓[P x \ V1 = ∅. The main idea of the algorithm is that if there exists a minimal St-serie decomposition (X1 , Z, X2) of P then there exists i ∈ [|V (P )| − 1] such that X2 = ∪ij=1 Li . Thus to verify that an order is St-serie decomposable or not, it is sufficient to check, for every i ∈ [|V (P )| − 1], if the sets X2 = ∪ij=1 Li , X1 = ↓∩P X2 and Z = V (P ) \ (X1 ∪ X2 ) induce a St-serie decomposition of P . Its good time complexity is obtained by a carefull writing of the macros Initialization, Create, and Update. Theorem 6 For a given connected order P , we can decide if it is St-serie decomposable in a time complexity in O(|V (P )| + | <P |). Moreover a St-serie 91 decomposition can be given in the same time complexity. 4 Linear time recognition of bisplit graphs In [2], A. Brandstädt et al define an undirected graph G = (V, E) to be bisplit if V can be partitionned into three stables sets X, Y, Z such that Y ∪Z induces a complete bipartite graph. Their interest in this class was a generalization of split graphs by replacing the clique set by a complete bipartite graph. Using the linear time algorithm for transitive orientation of R.M. McConnell and J.P. Spinrad [3], they obtain an O(mn) time recognition algorithm for bisplit graphs. Recently, A. Abueida and R. Sritharan [1] gave a simple O(n2 ) recognition algorithm which avoids the utilization of the transitive orientation algorithm as a subroutine. Taking advantage of the notion of St-serie decomposition, we can propose an O(n+m) time recognition algorithm for bisplit graphs. For this purpose we use both the transitive orientation algorithm of R.M. McConnell and J.P. Spinrad, and the algorithm we have already presented in Section 3. As bipartite graphs are also bisplit, the only remaining case is the recognition of none bipartite bisplit graphs. Since bisplit graphs are clearly comparability graph, as already noticed by A. Brandstädt et al, the characterization of bisplit graphs in terms of orders was then a natural question: Theorem 7 A not bipartite graph G is bisplit if and only if it exists an order P of height 2 such that G(P ) = G and that there exists a St-serie decomposition (X1 , Z, X2) of P such that both X1 and X2 are antichains of P . Notice that in the above theorem we can not replace the St-serie decomposition by a minimal one: the complete graph on three elements is obviously a biplist graph and for any minimal St-serie decomposition (X1 , Z, X2) of the three elements chain we have that either X1 or X2 is not antichain (but any St92 serie decomposition with each set being a singleton fulfills the condition of the theorem). Theorem 8 For an order P , we can determine if it exists a St-serie decomposition (X1 , Z, X2) where both X1 and X2 are antichains, in a time complexity in O(|V (P )| + | <P |). Moreover a such decomposition can be given in the same time complexity. Theorem 9 The recognition of bisplit graphs can be done in a time complexity in O(n+m) where n denotes the number of vertices and m the number of edges. References [1] Atif Abueida and R. Sritharan. A note on the recognition of bisplit graphs. Discrete Math., 306(17):2108–2110, 2006. [2] Andreas Brandstädt, Peter L. Hammer, Van Bang Le, and Vadim V. Lozin. Bisplit graphs. Discrete Math., 299(1-3):11–32, 2005. [3] Ross M. McConnell and Jeremy P. Spinrad. Modular decomposition and transitive orientation. Discrete Math., 201(1-3):189–241, 1999. [4] Bernd S. W. Schröder. Ordered sets. Birkhäuser Boston Inc., Boston, MA, 2003. An introduction. [5] Robert E. Tarjan. Decomposition by clique separators. 55(2):221–232, 1985. Discrete Math., [6] S. H. Whitesides. An algorithm for finding clique cut-sets. Inform. Process. Lett., 12(1):31–32, 1981. 93 94 The number of vertices whose out-arcs are pancyclic in 2-strong tournaments Ruijuan Li a, Shengjia Li a and Jinfeng Feng b a School of Mathematical Sciences, Shanxi University 030006 Taiyuan, P. R. China b Lehrstuhl C für Mathematik, RWTH Aachen University 52056 Aachen, Germany Key words: Tournament, Out-arc, Pancyclic Abstract An arc going out from a vertex x in a digraph is called an out-arc of x. A tournament is an orientation of the edges of a complete graph. An arc or a vertex in a digraph D with n ≥ 3 vertices is said to be pancyclic if it is contained in a k-cycle for all k satisfying 3 ≤ k ≤ n. Yao, Guo and Zhang [4] proved that every strong tournament contains a vertex x such that all out-arcs of x are pancyclic. Recently, Yeo [5] proved that each 3-strong tournament contains two such vertices. We confirm that Yeo’s result also is true for 2-strong tournaments. Our proof implies a polynomial algorithm to find two such vertices. 1 Introduction In [2], Goldberg and Moon proved that every s-arc-strong tournament has at least s distinct Hamiltonian cycles. Thomassen ([3]) confirmed that every strong tournament contains a vertex x such that each out-arc of x is contained in a Hamiltonian cycle. In 2000, Yao, Guo and Zhang improved the result of Thomassen as follows: Theorem 1 ([4]) Every strong tournament T with n ≥ 3 vertices contains a vertex u such that all out-arcs of u are pancyclic. Remark 2 In Theorem 1, if the vertices of T are labeled u1 , u2 , . . . , un with d+ (u1 ) ≤ d+ (u2 ) ≤ . . . ≤ d+ (un ), then, by Lemma 2.2 and Theorem 3.1 in [4], one of {u1 , u2} can be chosen as the vertex u with the property. It is easy to 95 check that d+ (u) ≤  n−1 2  . Therefore, every strong tournament T with n ≥ 3 vertices contains a vertex u with d+ (u) ≤ pancyclic.  n−1 2  such that all out-arcs of u are For 3-strong tournaments, Yeo proved the following: Theorem 3 ([5]) Every 3-strong tournament has two distinct vertices x and y, such that all arcs out of x and all arcs out of y are pancyclic. Furthermore, x and y can be chosen, such that x → y and d+ (x) ≤ d+ (y). In this paper, we prove that every 2-strong tournament contains at least two vertices v1 , v2 such that all arcs out of vi for i = 1, 2 are pancyclic. Our proof yields a polynomial algorithm to find such two vertices. 2 Path-contracting We will use the operation of path-contracting introduced in [1]. Let x and y be two distinct vertices of D and let P be an (x, y)-path in D. We say that H is obtained from D by contracting P to w, if the following holds: V (H) = (V (D)\V (P ))∪{w}, where w is a new vertex. Furthermore, NH+ (w) = ND+ (y) ∩ (V (D) \ V (P )), NH− (w) = ND− (x) ∩ (V (D) \ V (P )) and an arc with both end-vertices in (V (D)\V (P )) belongs to H if and only if it belongs to D. Note that uwv is a path in H, if and only if uP v is a path in D. Analogously, if there exists a k-cycle containing w in H, then there is a (k − 1 + |V (P )|)-cycle containing P in D. By using the operation of path-contracting, Yeo proved Lemma 4 ([5]) Let T be a 2-strong tournament containing an arc e = xy such that d+ (y) ≥ d+ (x). Then e is pancyclic in T . 3 Main result and sketch of proof Theorem 5 Each 2-strong tournament T with n vertices contains at least 2 vertices v1 , v2 such that all out-arcs of vi are pancyclic for i = 1, 2. Sketch of proof. By Theorem 3, we consider only the case when σ(T ) = 2. Let M contain all vertices in T , which have minimum out-degree. All out-arcs of every vertex in M are pancyclic by Lemma 4. If |M| ≥ 2, then we are done. 96 So, assume that |M| = 1 and denote M = {v1 }. Note that d+ (v) > d+ (v1 ) ≥ 2 for all v = v1 . In the following two cases, we successively choose v2 and can prove that either all out-arcs of v2 are pancyclic or there exists another vertex apart from v1 , v2 whose out-arcs are pancyclic. Case 1: σ(T − v1 ) = 1. Let S = {v1 , x} be a minimum separating set of T and let T1 , T2 , . . . , Tt (t ≥ 2) be the strong components of T − S. By Theorem 1, the subtournament Tt contains a vertex, say v2 , whose out-arcs are pancyclic in Tt . By Remark 2, |V (Tt )|−1 we can choose v2 such that d+ ⌋. Tt (v2 ) ≤ ⌊ 2 Case 2: σ(T − v1 ) = 2. Let v2 be the vertex, which has minimum out-degree among all vertices in N + (v1 ). In this case, we use the operation of path-contracting to show that v2 is another vertex whose out-arcs are pancyclic. References [1] J. Bang-Jensen and G. Gutin, Digraph: Theory, Algorithms and Application, Springer, London, 2000. [2] M. Goldberg and J.W. Moon, Cycles in k-strong tournaments, Pacific J. Math., 40 (1972), 89-96. [3] C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B, 28 (1980), 142-163. [4] T. Yao, Y. Guo and K. Zhang, Pancyclic out-arcs of a vertex in a tournament, Discrete Appl. Math., 99 (2000), 245-249. [5] A. Yeo, The number of pancyclic arcs in a k-strong tournament, J. Graph Theory, 50 (2005), 212-219. 97 98 A local tournament contains a vertex whose out-arcs are g-pancyclic Shengjia Li a, Wei Meng a and Yubao Guo b a School of Mathematical Sciences, Shanxi University 030006 Taiyuan, P. R. China b Lehrstuhl C für Mathematik, RWTH Aachen University 52056 Aachen, Germany Key words: Local tournaments, Out-arcs, Pancyclicity Abstract An arc going out from a vertex x in a digraph is called an out-arc of x. Thomassen [4] proved that every strong tournament contains a vertex x such that all out-arcs of x are contained in a Hamiltonian cycle. In 2000, Yao, Guo and Zhang [5] improved the result of Thomassen and confirmed that every strong tournament contains a vertex whose out-arcs are pancyclic. In this paper, we extend the result of Yao, Guo and Zhang to local tournaments and obtain a corresponding result. 1 Terminology and introduction Let D be a digraph. The out-neighbourhood (in-neighbourhood, respectively) of a vertex x is denoted by DN + (x) (DN − (x), respectively). An arc going out from a vertex x in a digraph is called an out-arc of x. If an arc in a digraph D with n ≥ 3 vertices is contained in a k-cycle for all k satisfying 3 ≤ k ≤ n, then it is said to be pancyclic. A digraph D is locally semicomplete, if DN + (x) and DN − (x) are both semicomplete for every vertex x of D. A locally semicomplete digraph containing no cycle of length 2 is called a local tournament. It is clear that every tournament is a local tournament, too. A local tournament is called a proper local tournament, if it contains at least two non-adjacent vertices. A digraph on n vertices is called a round digraph if we can label its vertices v0 , v1 , ..., vn−1 such that for each i, N + (vi ) = {vi+1 , ..., vi+d+ (vi ) } and 99 N − (vi ) = {vi−d− (vi ) , ..., vi−1 }, where the subscripts are taken modulo n. A locally semicomplete digraph D is round-decomposable, if there exists a round local tournament R on α ≥ 2 vertices such that D = R[D1 , D2 , ..., Dα ], where each Di is a strong semicomplete subdigraph of D for i = 1, 2, ..., α. We call R[D1 , D2 , ..., Dα ] a round decomposition of D. Let D be a strong local tournament with n vertices. The pseudo-girth g of D is defined as follows: If D is round-decomposable and it has the round decomposition D = R[D1 , D2 , ..., Dα ], then g = min{ n, max {gxi (R)} + 1 }, 1≤i≤α where xi is the vertex of R corresponding to Di and gxi (R) is the length of the shortest cycle containing xi in R; if D is not round-decomposable, then g = 3. In 1980, Thomassen [4] obtained the following result: Theorem 1 (Thomassen [4]) Every strong tournament contains a vertex x such that each out-arc of x is contained in a Hamiltonian cycle. In 2000, Yao, Guo and Zhang improved the result of Thomassen as follows: Theorem 2 (Yao, Guo and Zhang [5]) Every strong tournament with n ≥ 3 vertices contains a vertex u such that all out-arcs of u are pancyclic. It is an interesting problem whether Theorem 2 can be extended to local tournaments. In this talk, we give an affirmative answer to this problem and obtain a corresponding result. 2 Preliminaries First, the classification of locally semicomplete digraphs is as follows, which was obtained by Guo in 1995. Theorem 3 (Guo [3]) Let D be a connected locally semicomplete digraph. Then exactly one of the following possibilities holds. (1) D is round-decomposable with a unique round decomposition R[D1 , D2 , ..., Dα ], where R is a round local tournament on α ≥ 2 vertices and Di is a strong semicomplete digraph for i = 1, 2, ..., α; (2) D is not round-decomposable and not semicomplete and it has the structure as described in Theorem 5; (3) D is a not round-decomposable, semicomplete digraph. 100 For a connected, but not strongly connected locally semicomplete digraph D, we can obtain a unique sequence D1′ , D2′ , ..., Dr′ as follows which is called the semicomplete decomposition of D. Theorem 4 (Guo and Volkmann [2]) If D is a connected locally semicomplete digraph that is not strong, then D can be decomposed into r ≥ 2 semicomplete subdigraphs D1′ , D2′ , .., Dr′ such that the following holds: (1) D1′ is the terminal component of D and Di′ consists of some strong components of D for i ≥ 2; ′ (2) Di+1 dominates the initial component of Di′ and there exists no arc from ′ ′ Di to Di+1 for i = 1, 2, ..., r − 1; (3) If r ≥ 3, then there is no arc between Di′ and Dj′ for i, j satisfying |j −i| ≥ 2. Based on Theorem 4, Guo [3] gave a characterization of the locally semicomplete digraphs which are not semicomplete and not round-decomposable. Theorem 5 (Guo [3]) Let D be a strong locally semicomplete digraph which is not semicomplete. Then D is not round-decomposable if and only if the following conditions are satisfied: (1) There is a minimal separating set S such that D − S is not semicomplete and for each such S, DS is semicomplete and the semicomplete decomposition of D − S has exactly three components D1′ , D2′ , D3′ ; (2) There are integers α, β, µ, ν with λ ≤ α ≤ β ≤ p − 1 and p + 1 ≤ µ ≤ ν ≤ p + q such that N − (Dα ) ∩ V (Dµ ) = ∅ and N + (Dα ) ∩ V (Dν ) = ∅, or N − (Dµ ) ∩ V (Dα ) = ∅ and N + (Dµ ) ∩ V (Dβ ) = ∅, where D1 , D2 , ..., Dp and Dp+1 , ..., Dp+q are the strong decompositions of D − S and DS, respectively, and Dλ is the initial component of D2′ . 3 Main results A local tournament is either a tournament or a proper local tournament, and if it is a tournament, then the existence of a vertex whose out-arcs are pancyclic is ensured by Theorem 2. So we only need to consider proper local tournaments. According to the classification of local tournaments (Theorem 3), we have to consider whether they are round-decomposable or not rounddecomposable. Based on using Theorem 5 and Theorem 4, we obtain the following main result. 101 Theorem 6 Let D be a strong proper local tournament with n vertices, then the following holds: (1) If D is round-decomposable, then D contains a vertex u whose out-arcs are g-pancyclic, unless n ≥ 6 is even and D is isomorphic to Rn2 , where Rn2 is a 2-regular, round local tournament on n vertices and g is the pseudo-girth of D; (2) If D is not round-decomposable, then D contains a vertex u such that all out-arcs of u are pancyclic. References [1] M.K. Goldberg and J.W. Moon. Cycles in k-strong tournaments. Pacific J. Math. 40 (1972), 89-96. [2] Y. Guo and L. Volkmann. Connectivity properties of semicomplete digraphs. J. Graph Theory 18 (1994), 269–280. [3] Y. Guo. Locally Semicomplete Digraphs. Aachener Beiträge zur Mathematik 13 (Eds.H.H.Bock, H.Th.Jongen and W.Plesken), Doktorarbeit, RWTH Aachen, Augustinus Buchhandlung, Aachen (1995), 92 pp. [4] C. Thomassen. Hamiltonian-connected tournaments. J. Combin. Theroy Ser. B 28 (1980), 142–163. [5] T. Yao, Y. Guo and K. Zhang. Pancyclic out-arcs of a vertex in tournaments. Discrete Appl. Math. 99 (2000), 245–249. 102 A useful characterization of the feasible region of binary linear programs Leo Liberti 1 Key words: Feasible region, facet, rounding, binary linear program. 1 Introduction Given a Binary Linear Programming (BLP) problem in the following general form:     minx cx   s.t. Ax ≤ b x∈  [P ] (1)    n   {0, 1} ,  (where x are the decision variables, c is a rational cost n-vector, A is a rational m × n matrix, and b is a rational m-vector), the convex hull is the convex combination of all feasible integral points; its importance lies in the fact that the relaxed solution of the continuous relaxation of (1) subject to the convex hull of all its feasible integral points is integer. In view of providing an explicit representation of the convex hull by listing all the facets, it is interesting to describe the integral feasible region in terms of interior points, i.e. hypercube vertices which are feasible in (1) and such that all their adjacent hypercube vertices are also feasible in (1) and exterior points, for which there is at least one infeasible adjacent hypercube vertex. Whereas interior points belong to trivial facets of the convex hull (i.e. those facets which are also hypercube facets), exterior points define all the non-trivial facets. In this work we use a particular type of rounding along the hypercube edges (called flattening) to derive all exterior points of the feasible region of BLPs. We also show how to exploit this characterization to derive practically useful valid inequalities passing through hypercube vertices, and their relation to Balas’ intersection cuts [1]. Other works in the literature which are closely related to this topic are geometric [3] and canonical [2] cuts; both of these also pass through hypercube vertices, and therefore also identify exterior points. 1 Email address: liberti@lix.polytechnique.fr (Leo Liberti). LIX, École Polytechnique, F-91128 Palaiseau, France 103 The main idea in this work is that if an intersection point p between a hyperplane Ai x = bi (arising from the inequality Ai x ≤ bi of the relaxed feasible polyhedron) with the edge segments of the unit hypercube is not integral, then it has a unique fractional component. The two integral neighbouring hypercube points x1 , x2 are then separated by Ai x = bi ; assuming Ai x1 ≤ bi and Ai x2 > bi , and supposing that x1 is feasible in (1), we “flatten” the constraint Ai x ≤ bi in the direction of the feasible point x1 . Flattening inequalities are designed to intersect feasible integral points, hence they are likely to provide fast convergence for a cutting plane algorithm whenever the current relaxed solution is near a hypercube edge; however, because they are not guaranteed to be valid cuts, they need to be paired with general-purpose valid cuts separating the current relaxed solution, such as intersection cuts [1]. Let C n = (V, E) be the graph structure of the unit hypercube in n dimensions, and ι : V → {0, 1}n be the (invertible) map sending each vertex of the hypercube graph into the corresponding unit hypercube vertex in Euclidean space. We denote the set of adjacent vertices of v as δ(v). Given distinct x, y ∈ Rn we let [x, y] be the closed segment joining x, y ((x, y) is an open segment, (x, y] and [x, y) are semi-closed segments). For each {u, v} ∈ E we let [u, v] = [ι(u), ι(v)], and [u, v] be the line containing [u, v]. We denote by H n = {ι(v) | v ∈ V } and  by H̄ n = {u,v}∈E [u, v]. Given a set T ∈ Rn of n linearly independent points, we let (π(T ), π0 (T )) ∈ Rn+1 be a vector (π1 , . . . , πn , π0 ) such that πx = π0 is the hyperplane passing through all the points in T . Given y ∈ aff(T ), let (π(T, y), π0(T, y)) ∈ Rn+1 be such that πx = π0 for all x ∈ T and πy > π0 . For all j ∈ {1, . . . , n} we denote by ej the j-th unit coordinate direction vector  (01 , . . . , 1j , . . . , 0n ), and let e = nj=1 ej be the vector with all entries set to 1. Let F = {x ∈ {0, 1}n | Ax ≤ b} be the feasible region of problem P , which we assume to be non-empty, and F̄ = {x ∈ [0, 1]n | Ax ≤ b} its continuous relaxation. The continuous relaxation P̄ of P is the problem min{cx | x ∈ F̄ }. Let F ◦ = {x ∈ F |δ(ι−1 (x)) ∈ F } be the integral interior of F , namely the set of hypercube points feasible in P such that their n adjacent points in C n are also feasible in P . For all i ∈ {1, . . . , m} let Ai be the i-th row of A, so that Ai x ≤ bi is the i-th problem constraint; let Ri = {x ∈ Rn | Ai x = bi } and R̄i = {x ∈ Rn | Ai x ≤ bi }. Given a solution x′ of P̄ , let I(x′ ) be the set of active constraint indices. 2 The flattening operator For i ≤ m and {u, v} ∈ E, we consider the set Niuv = Ri ∩ [u, v]. The following facts hold: (1) Niuv is either a single point, or empty, or the whole segment [u, v]. (2) If |Niuv | = 1, Ri is a separating hyperplane for the singleton sets {ι(u)}, 104 {ι(v)}; furthermore, Ai ι(u) ≤ bi ⇔ (ι(v) − ι(u))Ai > 0, and Ai ι(u) > bi ⇔ (ι(v) − ι(u))Ai < 0. (3) If Niuv = [u, v], then (ι(v) − ι(u))Ai = 0 and both ι(u), ι(v) are in R̄i . For all i ≤ m let Ni =  Niuv . {u,v}∈E Lemma 1 For all i ≤ m and p ∈ Ni , there exists at most one component of p that is fractional. For i ≤ m and p ∈ Ni , we denote by f (p) the unique component index of p that might be fractional. Define: ⌊p⌋ = ⌈p⌉ =    (p1 , . . . , ⌊pf (p) ⌋, . . . , pn )   (p , . . . , 0, . . . , p ) 1 n    (p1 , . . . , ⌈pf (p) ⌉, . . . , pn )   (p 1 , . . . , 1, . . . , pn ) if 0 < pf (p) < 1 otherwise if 0 < pf (p) < 1 otherwise For {u, v} ∈ E we define u < v if there is j ≤ n such that (ι(v) − ι(u)) = ej , and u > v if there is j ≤ n such that (ι(v) − ι(u)) = −ej . Assuming u < v then it is straightforward to show that ⌊p⌋ = ι(u) and ⌈p⌉ = ι(v); furthermore, (⌈p⌉ − ⌊p⌋) = ef (p) . To each p ∈ Ni (i ≤ m) we associate the “closest” feasible integral point. For i ≤ m and p ∈ Ni we define the flattening of p as: Φ(p) =     ⌊p⌋    ⌈p⌉       {⌊p⌋, ⌈p⌉} if ⌊p⌋ ∈ R̄i , ⌈p⌉ ∈ R̄i if ⌈p⌉ ∈ R̄i , ⌊p⌋ ∈ R̄i if ⌊p⌋, ⌈p⌉ ∈ R̄i Let N = i≤m Ni be the set of all intersection points of the hyperplanes defining the problem constraints with the unit hypercube edges. The flattening of N is Φ(N) = {Φ(p) | p ∈ N} ∩ F̄ .  Theorem 2 {F ◦ , Φ(N)} is a partition of F . The main limitation of Thm. 2 is that for any given i ≤ n and {u, v} ∈ E, |Niuv | is generally not polynomial in n, but depends on the number of edges  in the unit hypercube, which is nd=1 nd d. 105 We recall that a facet of P is a hyperplane πx = π0 such that dim aff({x | πx = π0 } ∩ conv(F )) = n. The following results characterizes the extent to which flattened points can be used to derive facets of conv(F ). Theorem 3 Assume dim aff(F ) = n. Let W ⊆ Φ(N) such that (a) |W | = n,  π0 (W )). Then π(W )x = π0 (W ) is a facet of conv(F ). (b) ∀ w ∈ F ◦ (π(W )w = In practice, Thm. 3 cannot really be used to derive facets because testing condition (b) would yield exponential time complexity. We can, however, derive some cutting planes by flattening just one point at a time. Proposition 4 For i ≤ m and p ∈ Ni such that p is not integral, let q1 , . . . , qn−1 be the vertices adjacent to p in the (n − 1)-dimensional polyhedron Ri ∩ H̄ n , and W = {Φ(p), q1 , . . . , qn−1 }. Then π(W, p)x ≤ π0 (W, p) is a cutting plane for P . The cutting planes described in Prop. 4 are called flattening inequalities. Their most interesting feature is that they always pass through a polyhedron vertex. It is reasonable to expect that they should contribute to a faster convergence of cutting planes type algorithms by accelerating the identification of the optimal integral solution specially towards the end of the search (when the relaxed optima are expected to be nearer hypercube edges). 3 Conclusion In this paper we discussed a characterization of the feasible region of Binary Linear Programming problems in terms of interior and exterior points, and showed that this characterization is useful to derive some cutting planes. References [1] E. Balas. Intersection cuts — a new type of cutting planes for integer programming. Operations Research, 19(1):19–39, 1971. [2] E. Balas and R. Jeroslow. Canonical cuts on the unit hypercube. SIAM Journal on Applied Mathematics, 23(1):61–69, 1972. [3] N. Maculan, E.M. Macambira, and C.C. de Souza. Geometrical cuts for 0-1 integer programming. Technical Report IC-02-006, Instituto de Computação, Universidade Estadual de Campinas, July 2002. 106 On the maximum independent set problem in subclasses of planar and more general graphs Vadim Lozin a Martin Milanič b a RUTCOR, Rutgers University, 640 Bartholomew Rd, Piscataway NJ, 08854-8003, USA. E-mail: lozin@rutcor.rutgers.edu b RUTCOR, Rutgers University, 640 Bartholomew Rd, Piscataway NJ, 08854-8003, USA. E-mail: mmilanic@rutcor.rutgers.edu Key words: Maximum independent set problem, Polynomial-time algorithm 1 Introduction An independent set in a graph G is a subset of pairwise non-adjacent vertices. The maximum independent set problem (IS for short) is that of finding in a graph an independent set of maximum cardinality. If each vertex of G is assigned a positive integer, the weight of the vertex, then we say that G is a weighted graph. The maximum weight independent set problem (WIS for short) consists in finding in a weighted graph an independent set of maximum total weight. The IS problem is known to be NP-hard in general. Moreover, it remains NP-hard even under substantial restrictions, for instance, for trianglefree graphs [10], K1,4 -free graphs [8], and planar graphs of degree at most three, as shown by Garey, Johnson and Stockmeyer [6]. A related hardness result was given by Alekseev [1], who proved the following theorem. Theorem 1 ([1]) Let X be the class of graphs defined by a set F of forbidden induced subgraphs. If F is finite and contains no graph whose every connected component is of the form Si,j,k (see Figure 1), then the maximum independent set problem is NP-hard in X. In this paper, we address complexity issues for the IS problem in hereditary subclasses of planar graphs. First, we present a common strengthening of the results of Garey, Johnson and Stockmeyer, and Alekseev, by revealing restrictions on the set F of forbidden induced subgraphs, under which the problem remains NP-hard (Theorem 2). Then, by violating these restrictions, we obtain several polynomial-time results for the maximum independent set problem in subclasses of planar graphs (Theorem 3). Our polynomial 107 ri r❵ i−1 ❵❵ r ❝ 2 r1 r 1 r 2r r✟1r✟❍❍ ❵ ❵ ❵ k−1 ❵ ❵ ❵ 2 r r ❍rk r✟j−1 ❝ j ❝ 1 ❝ 2 i ❝ ❝ ❵ ❵ ❵ ❝ ❝ ❝ Fig. 1. Graphs Si,j,k (left) and Hi (right) results can also be extended to a more general setting. All graphs considered are finite, simple and undirected. 2 A hardness result Let Ci and Hi denote the cycle of length i and the graph on the right of Figure 1, respectively. We associate to every graph G a parameter κ(G): the minimum value of i ≥ 1 such that G contains an induced copy of either Ci or Hi . If G is an acyclic graph with no induced graphs of the form Hi , we let κ(G) = ∞. For a (possibly infinite) nonempty set of graphs F , we define κ(F ) = sup { κ(G) : G ∈ F } . With these definitions in mind, one can use the result of Garey, Johnson and Stockmeyer about the NP-hardness of the IS problem in planar graphs of degree at most three [6], and the reduction typically used for the IS problem (see e.g. [10]), to derive the following hardness result. For an arbitrary set of graphs Z, we denote by Z3 the set of graphs of vertex degree at most 3 in Z. Theorem 2 Let F be a set of graphs and X the class of F -free planar graphs. If κ(F3 ) < ∞, then the maximum independent set problem is NP-hard in X3 . 3 Polynomial results Let us now assume that X is the class of F -free planar graphs, for some set F of forbidden induced subgraphs. Unless P = NP, Theorem 2 suggests that the IS problem is solvable in polynomial time for graphs in X only if (i) F contains graphs of maximum degree at most 3 with arbitrarily large girth (i.e., the size of a smallest cycle) or (ii) F contains graphs of maximum degree at most 3 with arbitrarily large size of a smallest induced copy of Hi , or (iii) F contains a graph every connected component of which has the form Si,j,k . 108 In accordance with these observations, we prove the following result. Theorem 3 The maximum independent set problem admits the following solutions: (i) a linear-time algorithm for weighted F -free planar graphs with F = {Ck , Ck+1 , Ck+2, . . .} for any k ≥ 3; (ii) a polynomial-time 2-approximation algorithm for weighted F -free planar graphs with F = {Hk , Hk+1, . . . } for any k ≥ 1; (iii) a polynomial-time algorithm for F -free planar graphs with F = {S1,2,k } for any k ≥ 2. Part (i) is obtained by showing that the treewidth of graphs in such classes is bounded above by a constant, thus giving linear-time solutions to many other optimization problems too [2]. This boundedness of treewidth can be derived from an analogous result for graphs of bounded vertex degree [3] and the following observation which follows from a result of Demaine and Hajiaghayi [4]. Lemma 1 Let X be a subclass of planar graphs, and let Y denote the class of all graphs that can be obtained from graphs in X by applying a sequence of (zero or more) edge contractions and vertex deletions. If there is a function f such that tw(G) ≤ f (∆(G)), for all G ∈ Y , then there is an integer N such that treewidth of graphs in X is bounded above by N. Part (ii) is derived by finding in polynomial time a partition of the vertex set of the input graph into two parts V1 and V2 that induce a claw-free graph and a graph of bounded treewidth, respectively. The claw is the graph S1,1,1 , and polynomial-time solvability of the WIS problem in claw-free graphs is a classic result [8,9]. The existence of an approximation algorithm as in (ii) also follows from a PTAS for the WIS problem in any minor-closed family [5]. Nevertheless, we believe that it is worth mentioning our approach, mainly because it is so simple: one part of the bipartition above is given by V2 = {v ∈ V (G) : v is the center of an induced claw in G}. The derivation of part (iii) of the above theorem is much more involved. It consists of two general steps. First, we reduce the problem to S1,2,2 -free planar graphs, which is again done by means of bounding of the treewidth. Then we develop a direct solution to the IS problem in S1,2,2 -free planar graphs by combining the method of augmenting graphs [7] with the decomposition by clique separators [11]. The omission of the word “planar” from Theorem 3 results in graph classes for which the complexity status of the maximum independent set problem is unknown, with the two exceptions of F = {C3 , C4 , . . .} (forests) and 109 F = {C4 , C5 , . . .} (chordal graphs). However, Theorem 3 admits the following generalization. An apex graph is a graph that contains a vertex whose removal leaves a planar graph. For example, both minimal obstructions to planarity, K5 and K3,3 , are apex graphs. The results of Theorem 3 remain valid if one replaces the planarity condition with the condition “excludes a fixed apex graph H as a minor,” where H is arbitrary in parts (i) and (ii) and H = K3,3 in part (iii). References [1] V.E. Alekseev, On the number of maximal independent sets in graphs from hereditary classes, Combinatorial-algebraic methods in discrete optimization, University of Nizhny Novgorod, (1991) 5–8 (in Russian). [2] S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math. 23 (1989), 11–24. [3] H.L. Bodlaender and D.M. Thilikos, Treewidth for graphs with small chordality, Discrete Appl. Math. 79 (1997) 45–61. [4] E.D. Demaine and M.T. Hajiaghayi, Diameter and treewidth in minorclosed graph families, revisited, Algorithmica 40 (2004), 211–215. [5] E.D. Demaine, M.T. Hajiaghayi and K.-i. Kawarabayashi, Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring, Proceedings of the 46th FOCS, 2005, Pittsburgh, PA, October 23-25, 2005, 637– 646. [6] M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NPcomplete graph problems, Theoret. Comput. Science, 1 (1976) 237–267. [7] V.V. Lozin and M. Milanič, On finding augmenting graphs, RUTCOR Research Report 30-2005, Rutgers University, 2005 (available online at http://rutcor.rutgers.edu/pub/rrr/reports2005/38 2005.pdf). [8] G.J. Minty, On maximal independent sets of vertices in claw-free graphs. J. Combin. Theory Ser. B, 28 (1980) 284–304. [9] D. Nakamura and A. Tamura, A revision of Minty’s algorithm for finding a maximum weight stable set of a claw-free graph, J. Oper. Res. Soc. Japan 44 (2001) 194–204. [10] S. Poljak, A note on stable sets and Comment. Math. Univ. Carolinae 15 (1974) 307–309. coloring of graphs, [11] R.E. Tarjan, Decomposition by clique separators. Discrete Math., 55 (1985) 221–232. 110 Multiobjective Hierarchical Control of Time-Discrete Systems and Determining Stackelberg Strategies Dmitrii Lozovanu, a,∗,1 Stefan Pickl b a Institute of Mathematics and Computer Science, Academy of Sciences, Academy str., 5, Chisinau, MD–2028, Moldova b Institut fur Angewandte Systemwissenschaften und Wirtschaftsinformatik, Fakultat fur Informatik, Universitat der Bundeswehr, Munchen Abstract We consider a time-discrete system with the finite set of states, where the starting and the final states of the dynamical system are given. The dynamics of the system is controlled by p players and the order of fixing of vectors of control parameters of players in the control process is given for an arbitrary state at every moment of time. Each player intens to minimize his integral-time cost of the system’s passage by a trajectory from the starting state to the final state. We are seeking for a Stackelberg solution of the considered hierarchical control problem and develop dynamic programming technique for finding such a solution, which is based on the time-expanded network method. Key words: Time-Discrete System, Hierarchical Control, Stackelberg Solution, Time-Expanded Network Method, Dynamic Programming 1 Introduction and Problem Formulation In this paper we study the following multiobjective hierarchical discrete control problem. ∗ Dmitrii LOZOVANU Email address: lozovanu@math.md, spickl@informatik.unibw-muenchen.de (Stefan Pickl). 1 Supported by CRDF-MRDA 008 award MOM2-3049-CS-03 111 Let be given the time-discrete system L with finite set of states X ⊆ Rn . At every moment of time t = 0, 1, 2, . . . the state of dynamical system is x(t) ∈ X. Two states x0 and xf are given in X, where x0 = x(0) represents the starting point of the dynamical system L and xf is the state in which the system must be brought, i.e. xf is the final state of L. We assume that the system should reach the final state xf at the moment of time T (xf ) such that T1 ≤ T (xf ) ≤ T2 , where T1 and T2 are given. The dynamics of the system is controlled by p players and it is described as follows x(t + 1) = gt (u1 (t), u2 (t), . . . , up (t)), t = 0, 1, 2, . . . , (1) x(0) = x0 (2) where i mi is the starting point of the dynamical system and u (t) ∈ represent the vector of control parameters of player i, for which the admissible sets Uti (x(t)) are given, i.e. (3) ui (t) ∈ Uti (x(t)), t = 0, 1, 2, . . . ; i = 1, p. We consider that in the control process (1)-(3) at every moment of time t for an arbitrary state x(t) ∈ X the players fix their vectors of control parameters successively one after another according to their numbers orders. Each player fixing his vectors of control parameters inform posterior players which vector of control parameters has been chosen at the given moment of time. So, we will use the concept of hierarchical control [1]. Let u1 (t), u2 (t), . . . , up (t) be an arbitrary fixed set of control parameters of players 1, 2, . . . , p. Then u1 (t), u2(t), . . . , up (t) generate a trajectory x(0), x(1), x(2), . . . which either passes through the final state xf or does not pass through xf . If this trajectory passes through the final state xf , then T (xf ) represents the time-moment when the state xf is reached. We denote by T (xf )−1 Fxi0 xf (u1 (t), u2 (t), . . . , up (t)) =  cit (x(t), gt (x(t), u1 (t), u2 (t), . . . , up (t))) i=0 the integral-time cost of the system’s passage from x0 to xf for the player i, i ∈ {1, 2, . . . , p}, if the vectors u1 (t), u2 (t), . . . , up(t) satisfy condition (3) and generate a trajectory x0 = x(0), x(1), x(2), . . ., x(T (xf )) = xf from x0 to xf such that T1 ≤ T (xf ) ≤ T2 ; otherwise we put Fxi0 xf (u1 (t), u2 (t), . . . , up (t)) = ∞. Note that cit (x(t), gt (x(t), u1 (t), u2 (t), . . . , up (t))) = cit (x(t), x(t+ 1)) represents the cost of the system’s passage from the state x(t) to the state x(t + 1) at the stage [t, t + 1] for the player i. 112 We consider the problem of finding the control u1∗ (t), u2∗ (t), . . . , up∗ (t), for which u1∗ (t) = u1 (t)∈U 1 ,u2 (t)∈R2 (u1 ),u3 (t)∈R3 (u1 ,u2 ),...,up (t)∈Rp (u1 ,u2 ,...,up−1 ) Fx20 xf (u1∗ (t), u2 (t), . . . , up (t)); argmin u2∗ (t) = u2 (t)∈R2 (u1∗ ),u3 (t)∈R3 (u1∗ ,u2 ),...,up (t)∈Rp (u1∗ ,u2 ,...,up−1 ) Fx30 xf (u1∗ (t), u2∗ (t), . . . , up (t)); argmin 3∗ u (t) = Fx10 xf (u1(t), u2 (t), . . . , up (t)); argmin u3 (t)∈R3 (u1∗ ,u2∗ ),u4 (t)∈R4 (u1∗ ,u2∗ ,u3 ),...,up (t)∈Rp (u1∗ ,u2∗ ,...,up−1 ) ............................................................... Fxp0 xf (u1∗ (t), u2∗ (t), . . . , up−1∗(t), up (t)); argmin up∗(t) = up (t)∈Rp (u1∗ ,u2∗ ,...,up−1∗ ) where Rk (u1 (t), u2 (t), . . . , uk−1(t)) represents the best response of player k when players 1, 2, . . . , k − 1 have already fixed vectors u1 (t), u2 (t), . . . , uk−1 (t), i.e. R2 (u1 ) = 1 u2 (t)∈U 2 ,u3 (t)∈R3 (u1 ,u2 ),...,up (t)∈Rp (u1 ,u2 ,...,up−1 ) Argmin 2 R3 (u , u ) = Fx20 xf (u1 (t), u2 (t), . . . , up (t)); Argmin Fx30 xf (u1(t), u2 (t), . . . , up (t)); u3 (t)∈U 3 ,u4 (t)∈R4 (u1 ,u2 ,u3 ),...,up (t)∈Rp (u1 ,u2 ,...,up−1 ) ............................................................... Rp (u1 , u2, . . . , up−1) = Ui =  Fxp0 xf (u1(t), u2 (t), . . . , up (t), up (t)); Argmin up (t)∈U p Uti (x(t)), t = 0, 1, 2, . . . ; i = 1, p. x(t) t So, we consider the problem of finding Stackelberg solution for the considered hierarchical discrete control problem. It is easy to observe that if the solution u1∗ (t), u2∗ (t), . . . , up (t) does not depend on order of fixing vectors of control parameters of players then we obtain the solution in the sense of Nash [2]. 113 2 The Main Results In order to develop dynamic programming technique for determining Stackelberg solution of the considered hierarchical control problem we use the time-expanded network method from [3]-[6]. We formulate the problem on T2 -partite dynamic network with T2 + 1 level sets of states corresponding to the time-moments 0, 1, 2, . . . , T2 + 1 in the control process. Two nodes x(t) and x(t + 1) in the network we connect with arc (x(t), x(t + 1)) if there exists control u1 (t), u2 (t), . . . , up (t) such that (1) holds. Then starting from the final level we calculate recursively the cost of the system’s passage from the states at the given moment of time to the final state taking into account the order of fixing vectors of control parameters of players and the costs of the system’s passage from the state of posterior levels to the final state. Efficient algorithms for determining the solution of the problem in the case of two and three players are proposed. If for the considered multiobjective control problem Nash equilibria exists then the proposed algorithms find this equilibrium. Additionally, we have formulated and studied the hierarchical control problem on an arbitrary network (which may contain cycles) and for different moments of time and different states the order of fixing vectors of control parameters of players may be different. Stationary and nonstationary cases of the hierarchical control problem have been analyzed and dynamic programming algorithms have been developed. Computational complexity of the proposed algorithms is discussed. References [1] Stackelberg, H. Marktform and Gleichgewicht. Springer-Verlag (1934) [2] Nash, J. Non Cooperative Games. Annals of Mathematics, 2, 286–295 (1951) [3] Lozovanu, D., Pickl, S. Nash Equilibria for Multiobjective Control of TimeDiscrete Systems and Polynomial-Time Algorithms for k-partite Networks. Central European Journal of Operation Research, 13(2), 127–146 (2005) [4] Lozovanu, D., Pickl, S. An approch for an algorithmic solution of discrete optimal control problems and their game-theoretical extension. Central European Journal of Operation Research, 16(4), 357–375 (2006) [5] Lozovanu, D., Pickl, S. Algorithms and the calculation of Nash Equilibria for multi-objective control of time-discrete systems and polynomial-time algorithms for dynamic c-games on networks. European Journal of Operational Research, 181(3), 199–216 (2007) (to appear) [6] Lozovanu, D., Pickl, S. Algorithms for solving multiobjective discrete control problems and dynamic c-games on networks. Discrete Applied Mathematics (2007) (accepted for publication) 114 Fast point-to-point shortest path queries on dynamic road networks with interval data Giacomo Nannicini 1,2 Philippe Baptiste 1 Daniel Krob 1 Leo Liberti 1 Key words: Shortest path, dynamic arc costs, PTAS, real-time data 1 Introduction Consider a weighted directed graph G = (V, A, c) (where c : A → R+ ) which represents a road network evaluated by travelling times (so the graph may not be Euclidean). Assuming that c changes every few minutes, that the cardinality of V is very large (several million vertices), and that each shortest path request must be answered in a few milliseconds, current technology does not allow us to find an exact optimum in the brief time window before the next change of the weight function. Such a situation occurs in fastest path computations for GPS enabled vehicles with real-time traffic information. Some practically efficient algorithms for graphs with fixed arc costs are [SS05,GKW05]; [MG04,MGD04,SP06] address uncertainty in arc costs but do not yield algorithms which perform within acceptable time frames. We assume some lower and upper bounding functions l, u : A → R for c are known. In this paper, we propose a Polynomial-Time Approximation Scheme (PTAS) for the Point-to-Point Shortest Path Problem (PPSPP). The stringent time constraints do not make exact algorithms an acceptable choice, yet a guarantee on the solution quality is desired. Our algorithm is based on Dijkstra-type searches performed on clusters of nodes; such clusters are precomputed in such a way as to give a bound on the solution performance, whilst Email addresses: giacomo.nannicini@v-trafic.com (Giacomo Nannicini), baptiste@lix.polytechnique.fr (Philippe Baptiste), dk@lix.polytechnique.fr (Daniel Krob), liberti@lix.polytechnique.fr (Leo Liberti). 1 LIX, École Polytechnique, 91128 Palaiseau, France 2 Mediamobile, 10 rue d’Oradour sur Glane, Paris, France 115 accelerating the search enough to be practically useful within the given time constraints. 2 Guarantee regions For s, t ∈ V we denote the set of all paths (s, . . . , t) from s to t by P (s, t) and the set of all shortest paths from s to t on a graph weighted by function f by Pf∗ (s, t) (the subscript f is omitted when f = c). Given U ⊆ V such that s, t ∈ U, let G[U] be the subgraph of G induced by U. The set of all paths between s and t in G[U] is denoted by P [U](s, t) and the set of all shortest paths between s and t in G[U] weighted by function f is denoted by Pf∗ [U](s, t); as before, we will write P ∗ [U](s, t) = Pc∗ [U](s, t). We naturally k−1 extend c to be defined on paths p = (v1 , . . . , vk ) by c(p) = i=1 c(vi , vi+1 ). Let Gl = (V, A, l), Gu = (V, A, u) be the graph G weighted by the lower and upper bounding functions l, u. For K > 1, s, t ∈ V and path p ∈ Pu∗ (s, t) (p is a shortest path from s to t in Gu ), we define a guarantee region Γst (K, p) = {v ∈ V |v ∈ p ∨ ∃ q ∈ P (s, t) (v ∈ q ∧ l(q) < K1 u(p))}. We can prove that such a set of nodes has the following approximation property: p∗ ∈ P ∗ (s, t) and q ∗ ∈ P ∗ [Γst (K, p)](s, t), we have c(q ∗ ) ≤ Kc(p∗ ). Although guarantee regions generated by shortest paths in Gu may not be the minimally-sized sets with this approximation property, it is possible to show that they are no worse than those generated by any other path. The trouble with the guarantee regions defined above is that building all guarantee regions for all node pairs in a very large graph is not a feasible task with current technology. We deal with this problem by covering V with clusters V1 , . . . , Vk ; however, not all coverings are useful for our approach. We will call a covering V1 , . . . , Vk of V valid if for all i ≤ k there are two selected (not necessarily distinct) vertices si , ti ∈ Vi such that for all other vertices v ∈ Vi there are paths p ∈ P (v, si), q ∈ P (ti , v) entirely contained in Vi . For all i ≤ k let σi = maxv∈Vi ,p∈Pu∗ (v,si ) c(p) and τi = maxv∈Vi ,p∈Pu∗ (ti ,v) c(p)) be the costs of the longest shortest path in Gu from v to si and respectively from ti to v over all v ∈ Vi . We can now extend guarantee regions to depend on a source and a destination cluster in a valid covering V1 , . . . , Vk of V . For K > 1, i = j ≤ k and a path p ∈ Pu∗ (si , tj ), we define the clustered guarantee region as ΓVi Vj (K, p) = {v ∈ V |v ∈ p ∪ Vi ∪ Vj ∨ ∃ q ∈ P (si , tj ) (v ∈ q ∧ l(q) < 1 (u(p) + σi + τj ))}. We can prove that such a set of nodes has the followK ing approximation property: for u ∈ Vi , v ∈ Vj (where i = j), p∗ ∈ P ∗ (u, v), q ∗ ∈ P ∗ [ΓVi Vj (K, p)](u, v), we have c(q ∗ ) ≤ Kc(p∗ ). Again, clustered guarantee regions may not be the minimally-sized sets having this approximation property, but they are no worse than those generated by any other path. 116 The definition of guarantee regions imply that the most expensive part of their (pre-processing) computation is finding all paths p ∈ P (s, t) s.t. l(p) < H. There exists in fact a polynomial algorithm that computes all nodes on a path with a total cost < H from a node s to a node t. Computing H itself is straightforward, since it requires knowing p∗ ∈ Pu∗ (s, t) and, in the clustered case, upper bounds to the cost of shortest paths in a small set of nodes. 3 Preliminary computational experiments We used a subgraph of France’s road network, roughly corresponding to Île-deFrance (i.e. Paris and surroundings), to validate our approach. This subgraph has roughly 300.000 vertices and 800.000 edges. We ran several bidirectional Dijkstra searches [SS05] on the full graph and on guarantee regions to assess the usefulness of our heuristic, with source and destination node chosen at random, and for each source-destination pair we repeated the query 5 times with arc costs generated at random with a uniform distribution each time; upper bounds on arc costs are between 5-10 times lower bounds. We recorded solution quality and CPU times in Table 1. For each value of K (first column), we indicate the average number D of nodes explored in bi-directional Dijkstra searches n the full graph, the average number R of nodes explored in bidirectional Dijkstra searches on the guarantee regions, the average percentage increase P of the approximated solution value with respect to the optimum (0% means that the approximated solution is optimal), the average CPU time savings C in percentage of the CPU times taken by the exact algorithm (0% means as slow as the exact algorithm). K 3 D 74559 R 74532 P 0% C 0% 4 74779 74219 0% 0% 5 74651 65126 0% 8.39% 6 74739 39282 0% 46.85% 7 74647 5609 0.07% 93.86% Table 1 Computational results on unclustered graph: mean values. To validate the clustered approach, we generated a valid covering of V , and then, for some random cluster pairs, compared the number of explored and settled nodes between a bidirectional Dijkstra search and a bidirectional Dijkstra search constrained to the guarantee regions, where source and destination node of Dijkstra’s search where chosen randomly in their respective cluster, performing 5 queries with arc costs generated at random for each sourcedestination pair. Results are reported in Table 2 (same column labels as Table 1); cluster size was set to 500 nodes. 117 K 6 D 74493 R 73262 P 0% C 0% 7 74605 66804 0% 5.83% 8 74129 56761 0% 20.35% 9 74436 34091 0.02% 54.26% 10 74494 13978 1.20% 82.05% Table 2 Computational results on clustered graph: mean values 4 Conclusion We have proposed a two-phase Polynomial Time Approximation Scheme for the Point-To-Point Shortest Path Problem, based on a pre-processing phase where we compute guarantee regions where we can restrict the search, and a query phase where we answer point-to-point queries. This was motivated by the need for finding short paths on large-scale graphs very quickly, where arc costs (representing travelling times given by real-time traffic data on a road network) may vary within a given range. The approach was successfully validated on a graph (of limited size) derived from the road network of the Île-de-France region, showing great computational time reduction in query resolution while still finding an optimal or near-optimal solution. In practice, real-time traffic data is only available for a relatively small subset of all arcs, so that knowing how to infer unknown traffic data from the existing data is also an issue. Although we do not target this particular aspect of the problem in this paper, future work will be undertaken on the matter. References [GKW05] A.V. Goldberg, H. Kaplan, and R.F. Werneck. Reach for A∗ : Efficient point-to-point shortest path algorithms. Technical Report MSR-TR-2005-132, Microsoft Research, 2005. [MG04] R. Montemanni and L. M. Gambardella. An exact algorithm for the robust shortest path problem with interval data. Computers & Operations Research, 31:1667–1680, 2004. [MGD04] R. Montemanni, L. M. Gambardella, and A. V. Donati. A branch and bound algorithm for the robust shortest path problem with interval data. Operations Research Letters, 32:225–232, 2004. [SP06] A. Sengupta and T. K. Pal. Solving the shortest path problem with interval arcs. Fuzzy Optimization and Decision Making, 5:71–89, 2006. [SS05] P. Sanders and D. Schultes. Highway hierarchies hasten exact shortest path queries. In G. Stølting Brodal and S. Leonardi, editors, ESA, volume 3669 of Lecture Notes in Computer Science, pages 568–579. Springer, 2005. 118 Decomposition Method for Project Scheduling with Adjacent Resources Jacob Jan Paulus and Johann Hurink Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Key words: Project Scheduling, Adjacent Resources 1 Introduction We develop a decomposition method for project scheduling problems with adjacent resources. Adjacent resources are resources for which the units assigned to an activity are required to be adjacent. Possible examples of adjacent resources are dry docks, shop floor spaces and assembly areas. We focus on the Time-Constrained Project Scheduling Problem (TCPSP) with 1-dimensional adjacent resources. The TCPSP with adjacent resources is defined as follows. We are given a set of activities, a set of renewable resources and a set of adjacent resources. Each activity is characterised by a release date, processing time and deadline, and has to be scheduled without preemption. The processing of activities is restricted by precedence relations. The adjacent resources are not required by a single activity but by groups of activities. As soon as an activity of such an activity group starts the assigned resource units are occupied, and they are not released before all activities of that group are completed. On top of that, the adjacent resource units that are assigned to an activity group have to be adjacent. For the renewable resources it is possible to hire additional capacity. The objective is to find a feasible assignment of the activity groups to the adjacent resources, and an activity schedule which minimises the cost of hiring additional capacity. A practical application from the ship building industry is one of many that motivates this study. In this real world problem the docks are 1-dimensional adjacent resources, and all activities related to building a single ship form an activity group. Clearly, the part of the dock assigned to one ship has to satisfy the adjacency requirement. As soon as the construction of a ship starts, the 119 assigned part of the dock is occupied until the construction is finished and the ship is removed from the dry dock. Removal or repositioning of a partially assembled ship is not possible. The capacity of the dry dock is fixed but the capacity of renewable resource can be increased, e.g. by hiring personnel. Although, both the concept of activity groups and adjacency requirement on resources have been treated in the literature, the combination has not. To the best of our knowledge we are the first to consider this problem. The main result of this paper is a solution approach which first searches for an allocation of the activity groups to the adjacent resources which allows a feasible solution for the overall problem. Using this assignment, we then constructs a schedule for the activities by incorporating the solution of the group assignment into an activity on node (AoN) network in such a way that the group structure and adjacent resources can be omitted. As a consequence we are left with an ordinary TCPSP. Since different assignments of activity groups imply adding different precedence relations, we investigate the effect of turning the search for a feasible assignment into an optimisation problem by adding an objective function to it. This allows for not just searching for any feasible assignment, but a “good” assignment. 2 Complexity and Existing Methods From the literature it is known that activity groups can be modelled with the concept of cumulative resources (see e.g [2]). However, the existing methods for project scheduling do not consider the adjacency requirement on resources. Employing an existing method on the TCPSP with adjacent resources, i.e. by modelling groups with cumulative resources, gives us start and completion times of the activities and groups, but no assignment of adjacent resource units to groups. Determining whether there exist an feasible assignment of adjacent resources given these start and completion times, is still an NP complete problem (see [1]). There is no guarantee the start and completion times found with an existing method, are such that there exists a feasible assignment. Therefore, we develop a decomposition method which first considers this feasibility issue. Since the assignment of groups to adjacent resources is an NP-complete problem, no heuristic is guaranteed to find a feasible assignment. The feasibility is essential in our problem, therefore, we use an exact approach in this first stage, i.e. an ILP formulation. 120 3 Decomposition Method The first step in our decomposition, is to determine an assignment of the groups to the adjacent resource units, and an ordering between the groups assigned to the same adjacent resource units. We call this the Group Assignment Problem (GAP). From an AoN representation of the project, we derive timing constraints on the groups such that a group assignment respecting these constraints allow for a feasible activity schedules. The order in which the groups are assigned to the same adjacent resource, implies additional precedence relations in the original problem. As a result, the adjacent resources do not have to be considered anymore. The resulting problem is a TCPSP without adjacent resources. The second step is to solve this resulting TCPSP, which can be done by employing existing methods. It is important to notice that one solution of the GAP can be much better than another one with respect to the resulting TCPSP. To distinguish between GAP solutions we propose a number of objective functions for the ILP that solves the GAP. Although, the computational tests show no dominance between the presented objective functions, all of them give much better results than an arbitrary feasible solution. 4 Conclusions With the presented decomposition approach, we can easily detect infeasibility of an instance by solving the GAP in the first step. In case an instance is feasible, the first step gives also a solutions for the group assignment and the order of the groups on the adjacent resources, which can be extended to an overall feasible solution in the second step. The test results show no clear dominance among the presented GAP objective functions, but adding any of the proposed objective functions improves the solution quality after step two a lot. Finding good solutions by one specific objective remains problematic, but by taking combinations of objective functions the quality of the generated schedules improves significantly. For future research it would be interesting to see under which conditions which objective function performs well, and to explore different methods to solve the GAP. Besides exploring a fixed objective, the presented decomposition can be the basis of a feedback between the GAP and the resulting TCPSP, where the outcome of the resulting TCPSP can influence the GAP objective before resolving. This may lead to a local search approach, where the weights and the different type of objectives of the GAP can be used as a solution space. 121 Adapting the weights can be seen as some sort of intensification phase and the change of the objective as some sort of diversification phase of the search process. To make such an approach successful, the computational time for solving the GAP has to be reduced (e.g. by not solving it to optimality) and intelligent ways of changing the weights based on the outcome of the TCPSP have to be developed. Summarising, the presented decomposition forms a first promising approach for the TCPSP with spatial resources and may form a good basis to develop better and more efficient methods. References [1] C.W. Duin and E. van der Sluis (2006), On the complexity of adjacent resource scheduling, Journal of Scheduling, 9, 49-62. [2] K. Neumann and C. Schwindt (2002), Project scheduling with inventory constraints, Mathematical Methods of Operations Research, 56, 513-533. 122 GrInvIn for Graph Theory Teaching and Research Adriaan Peeters, Kris Coolsaet Gunnar Brinkmann, Nicolas Van Cleemput and Veerle Fack TWI, Universiteit Gent, Krijgslaan 281 S9, B 9000 Gent, Adriaan.Peeters,Kris.Coolsaet,Gunnar.Brinkmann, Nicolas.VanCleemput,Veerle.Fack@UGent.be Key words: graph, graph invariant, conjecture, software framework 1 GrInvIn in a nutshell Various programs to support research in graph theory have been developed and successfully used, such as AGX/AGX2[1], Cabri-graph[2], Graffiti[3], Graffiti.pc[4], GRAPH[5], GraPHedron[6], LINK[7], and newGRAPH[8]. Some of them emphasize the manipulation of graphs and computation of invariants, others focus on (graph) conjecturing. As to the goal of GrInvIn we were most influenced by Graffiti.pc which was developed by Ermelinda DeLaVina. It was created for research in graph theory as well as for teaching graph theory by means of graph conjecturing. The GrInvIn framework provides the core functionality needed to implement an application for graph theory in general. It includes basic functionality to work with graphs, invariants, and conjectures. In addition to data structures and interfaces for these concepts, the framework also provides a basic graph editor, various invariant computing routines, and an intuitive graphical user interface. GrInvIn is still being developed and soon further functionality (such as graph generation programs) will be added. In order to guarantee portability, the interface and most of the subroutines are written in the highly portable programming language Java. Some parts that are performance critical and interact less with the operating system are written in C. 123 Fig. 1. Screenshot of the GrInvIn user interface. 1.1 User interface The GrInvIn user interface (Figure 1) makes use of the drag and drop concept to work with graphs and invariants. All GrInvIn objects are represented as icons. This user interface works in a way similar to that of the file manager found in most recent operating systems. To allow the user to work in their own native language, the entire user interface is internationalized. Translating the framework only involves providing a few language files, without any changes in the application source code. So far GrInvIn comes with language files in English and Dutch. 1.2 Teaching Graph Theory with GrInvIn GrInvIn comes with a conjecturing engine making conjectures, or rather guesses, based on a set of graphs given to the program by the user. These graphs can be input via a graph editor or a graph factory in case the graph belongs to one of several well known classes – such as complete graphs, cycles, paths, etc. The intention of this conjecturing engine is to be used for teaching graph theory to university or even high school students who make their first contacts with graph theory. 124 The basic strategy follows Fajtlovicz and DeLaVina who first applied these principles using their programs Graffiti and Graffiti.pc: The student starts with a set of invariants and a very small list of graphs – e.g. containing just K1 or K3 . He chooses an invariant for which he wants to determine (e.g. upper) bounds. Conjectures based on this small set of graphs can in general easily be shown to be false and often very small counterexamples exist. The student’s task is now to prove or disprove the conjecture given by the program. In case of a wrong conjecture the student has to prove that his counterexample is smallest possible and add that counterexample to the list. Then a new conjecture is computed for this larger list of graphs and the process is repeated. In case of a true conjecture, the student can prove the conjecture and try to find out whether the conjecture always gives a good bound for his invariant or whether the difference between the left hand and the right hand side can be arbitrarily large. Afterwards he can either input a graph where the difference between the two sides is very large or delete one of the invariants from the list to force the program to make new conjectures. Courses based on this strategy were given in Houston and later also in Bielefeld and Ghent. At the time of the conference, first courses of this kind will also have taken place already on the level of high school students in Ghent. Experience showed that this way to make contact with graph theory was very motivating for students. They were keen on learning about the new invariants occuring in the conjectures, because the conjecture connected the new invariant to their chosen invariant. Furthermore this way of learning has some air of discovery: There is always at least the chance to discover some new result – instead of just learning basic results studied already by generations of mathematicians. In Bielefeld (where Graffiti.pc was used) the effect on the students was very positive: the students did not only enjoy the course – afterwards they also asked for a normal graph theory course to learn more about graph theory and proof techniques in graph theory and half of the students later decided to write their thesis in this field. In our eyes graph theory is also ideal to teach logical thinking and argumentation to students already at high-school level and hopefully GrInvIn will turn out to be a good tool to do so. Unfortunately neither Graffiti nor Graffiti.pc are publically available so far. Graffiti was never meant to be distributed and – also due to lack of resources – Graffiti.pc develops also very slowly. Though being inspired by Graffiti.pc, GrInvIn is not based on Graffiti.pc, but was completely designed from scratch. The emphasis lies on the software engineering aspect with the aim to combine the functionality of Graffiti.pc with 125 a modern software design, a user friendly interface, and easy extensibility. 1.3 Content of the Talk In this talk we will give a short introduction into the use of GrInvIn and show an example scenario of a teaching session. References [1] Gilles Caporossi and Pierre Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system., Discrete Mathematics, 212, 1-2, 2000, p.29–44, http://dx.doi.org/10.1016/S0012-365X(99)00206-X. [2] Yves Carbonneaux, Jean-Marie Laborde, and Rafaı̈ Mourad Madani, CABRIGraph: A Tool for Research and Teaching in Graph Theory., Graph Drawing, 1995, p.123–126. [3] Ermelinda DeLaVina, Some History of the Development of Graffiti. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 69: Graphs and Discovery, 2005, p. 81–118. [4] Ermelinda DeLaVina, Graffiti.pc: A Variant of Graffiti, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 69: Graphs and Discovery, 2005, p.71–79. [5] Dragoš Cvetković and Slobodan Simić, Graph Theoretical Results Obtained by the Support of the Expert System ”Graph” – An Extended Survey, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 69: Graphs and Discovery, 2005, p.39–70. [6] Hadrien Melot, GraPHedron, see http://www.graphedron.net. [7] Jonathan W. Berry, Nathaniel Dean, Mark K. Goldberg, Gregory E. Shannon, and Steven Skiena, Graph Drawing and Manipulation with LINK., Graph Drawing, 1997, p.425–437. [8] Dragan Stevanović, Vladimir Brankov, Dragos̃ Cvetković, and Slobodan Simić, newGRAPH, see http://www.mi.sanu.ac.yu/newgraph/. 126 Edge cover by bipartite subgraphs Marie-Christine Plateau CEDRIC-CNAM, 292 rue Saint-Martin, Paris, France Leo Liberti LIX, École Polytechnique, F-91128 Palaiseau, France Laurent Alfandari ESSEC, Av. Bernard Hirsch, BP105 95021, Cergy Pontoise, France Key words: Bipartite subgraph, edge cover, integer linear programming, heuristics. 1 Introduction We consider the following optimization problem. Minimum Bipartite Graph Cover (MBGC). Given a connected undirected graph G = (V, E) without loops or parallel edges, find a family {Hk = (Ak , Bk , Ek ) | k ≤ m} of (not necessarily induced nor complete) con nected bipartite subgraphs of G such that E = k≤m Ek and m is minimum. Two related and reasonably well-studied problems are the Minimum Biclique Cover (MBC), where Hk are required to be bicliques (i.e. complete bipartite subgraphs) [6,5,2,1] and the Minimum Cut Cover (MCC), where Hk are cutsets, namely not required to be connected. Both problems are NPhard. To the best of our knowledge, whether the MBGC is NP-hard or not is currently unknown. Let G = (V, E) be an undirected graph. For v ∈ V , we denote by δ(v) the set of vertices u such that {v, u} ∈ E, and by δ̄(v) the set of edges e ∈ E adjacent to v. With respect to a set of edges F ⊂ E, δF (v) is the set of vertices adjacent to v using edges in F , and δ̄F (v) is the set of edges e ∈ F adjacent to v. Email addresses: mc.plateau@cnam.fr (Marie-Christine Plateau), liberti@lix.polytechnique.fr (Leo Liberti), alfandari@essec.fr (Laurent Alfandari). 127 2 Mathematical Formulation 2.1 Decision variables Let m̄ be an upper bound to m, for example m̄ = ⌈ n2 ⌉. We consider four sets of binary variables and a set of continuous multicommodity flow variables:   1 if the k-th bipartite subgraph is in the cover ∀k ≤ m̄ yk =  0 otherwise,     1 if edge (i, j) belongs to the k-th k bipartite subgraph ∀i ∈ V, j ∈ V, k ≤ m̄ eij =    0 otherwise,   1 if node i is in A k ∀i ∈ V, k ≤ m̄ aki =  0 otherwise,   1 if node i is in B k k ∀i ∈ V, k ≤ m̄ bi =  0 otherwise, ∀{i, j} ∈ E, k ≤ m̄, u ∈ V, v ∈ V : u = v fijuvk ∈ [0, 1] (1) (2) (3) (4) (5) where the continuous flow variables f identify a path connecting u and v in the k-th bipartite subgraph in order to ensure that bipartite subgraphs are connected, i.e., fijuvk = 1 if edge (i, j) ∈ E belongs to the path connecting u and v, 0 otherwise. 2.2 Objective function The objective is to minimize the number of subgraphs in the cover: min m̄  (6) yk k=1 2.3 Global covering constraints The global constraints linking all subgraphs are the edge covering constraints: ∀ (i, j) ∈ E : i < j m̄  k=1 128 ekij ≥ 1 (7) The remaining constraints are local constraints defining for each index k a valid subgraph (i.e., bipartite and connected). 2.4 Local Bipartite constraints Constraints (8),(9),(10),(11) ensure that subgraphs are bipartite. aki + bki ≤ y k ∀k = 1, . . . , m̄, ∀i ∈ V : (8) The above constraints have a double function: first, they ensure that every node in subgraph k is whether in Ak or B k but not both, second, they translate the logical constraints that if node i is in subgraph k then y k = 1. We have moreover: ∀ k = 1, . . . , m̄, (i, j) ∈ E : aki + akj ≤ 2 − ekij (9) ∀ k = 1, . . . , m̄, (i, j) ∈ E : bki + bkj ≤ 2 − ekij (10) ∀ k = 1, . . . , m̄, i ∈ V : aki + bki + akj + bkj ≥ 2ekij (11) These constraints ensure that if ekij = 1, then we have whether aki = 1, bki = 0, akj = 0, bkj = 1 or aki = 0, bki = 1, akj = 1, bkj = 0. This eliminates oddlength cycles as these cycles would have an edge ekij = 1 with aki = akj = 1 or bki = bkj = 1, so subgraph k is bipartite indeed. 2.5 Local connectivity constraints Define A = {(i, j), (j, i) : (i, j) ∈ E}, i.e. edges are transformed in two inversed arcs. The multicommodity flow constraints below ensure that each bipartite subgraph is connected: ∀ u ∈ V, v ∈ V, k ≤ m̄ : u = v  uvk fuj ≥ aku + bku + akv + bkv − 1 (12)  uvk fuj =0 (13)  uvk fiv ≥ aku + bku + akv + bkv − 1 (14)  uvk =0 fvj (15) j∈V :(i,j)∈A ∀ u ∈ V, v ∈ V, k ≤ m̄ : u = v i∈V :(i,u)∈A ∀ u ∈ V, v ∈ V, k ≤ m̄ : u = v i∈V :(i,v)∈A ∀ u ∈ V, v ∈ V, k ≤ m̄ : u = v j∈V :(v,j)∈A ∀ (u, v) ∈ V 2 , k ≤ m̄, j ∈ V : u = v, j = u, v  i∈V :(i,j)∈A 129 fijuvk =  l∈V :(j,l)∈A fjluvk (16) Constraints (12),(13),(14),(15) and (16) ensure that, if both nodes u and v are in subgraph k (in that case in constraints (12) and (14) the right-hand term aku +bku +akv +bkv −1 is one), then a single flow unit leaves u and finally (by constraints (16)) arrives at node v, hence defining a path connecting u and v. Constraints (13) and (15) are necessary to ensure the the flow is not composed of two disconnected cycles, one passing through u and the other one through v. We denote this integer program by (BGC). Lemma: (BGC) allows to find the optimal sequence {Hk = (Ak , Bk , Ek ) | k ≤ m}, that is one of the minimal subset of bipartite graph cover for the edges of G. Variants of the model introducing new variables are proposed and compared on a set of graph instances. 3 Efficient heuristic We mean to find a sequence {Hk | k ≤ m} of bipartite subgraphs of G such that the union of all their edges covers the edges E of G. Additionally, we would like to minimize m. We propose the simple but effective heuristic in Alg. 1; it finds a set of bipartite graphs Hk = (Ak , Bk , Ek ) with the required properties. Let m̄ be an upper bound to m. The heuristic works by constructing the k-th bipartite graph (Ak , Bk , Ek ) in the cover. It progressively selects the vertex v with highest star degree, disconnected from Ak but whose star intersects Bk ; v is added to Ak , its vertex star to Bk and its edge star to Ek . When no further vertices may be added to Ak , k is increased and the procedure is repeated with a smaller edge set E = E  Ek . When E = ∅, the cover is complete. The worst-case complexity of Algorithm 1 in a naive implementation is O(|V |3 |E|). Lemma 1 The sequence {Hk = (Ak , Bk , Ek ) | k ≤ m} found by Algorithm 1 is a bipartite graph cover for the edges of G. Proof. By inspection it is easy to see that at termination, the algorithm provides a bipartite graph cover for the edges of G. It remains to be shown that the algorithm terminates, namely that k never increases twice consecutively in Step (2) without |E| decreasing. This is easily shown as follows: at the iteration following the increase in k, we have Ak = Bk = ∅, whence U = Z = V in Step (1). Since U = ∅ and E = ∅ (since otherwise the algorithm would have 130 Algorithm 1 Fast heuristic for bipartite graph cover. Initialize all Hk to ∅ for k ≤ m̄. Let k = 1. while E = ∅ do if Ak = Bk = ∅ then Let U = Z = V . (1) else Let U = {v ∈ Z | ∀u ∈ Ak {u, v} ∈ E ∧ δE (v) ∩ Bk = ∅}. end if if U = ∅ then Set k ← k + 1. (2) else Let v ∈ U s.t. |δE (v)| is maximum. (3) Set Ak ← Ak ∪ {v}, Bk ← Bk ∪ δE (v), Z ← Z  ({v} ∪ δE (v)), Ek ← Ek ∪ δ̄E (v), E ← E  δ̄E (v). end if end while Let m = k. already terminated), there is a v ∈ U such that |δE (v)| ≥ 1 in Step (3). Thus |E| is decreased. ✷ The following table and corresponding CPU time vs. |V | plot illustrate the performance of Alg. 1 on a set of 10 randomly generated undirected graphs where each edge has unit cost and is generated with probability 0.5. All experiments were carried out on an Intel Core Duo 1.2GHz with 1.5 GB RAM running the Linux operating system. 18 "out" using 1:5 |V | 100 200 300 400 500 600 700 800 900 1000 B 20 32 45 56 68 81 91 101 111 121 user CPU time (s) 0.06 0.20 0.66 1.55 2.67 4.62 6.69 10.16 13.67 17.67 16 14 12 10 8 6 4 2 0 100 200 131 300 400 500 600 700 800 900 1000 References [1] G. Alexe, S. Alexe, Y. Crama, S. Foldes, P. Hammer, and B. Simeone. Consensus algorithms for the generation of all maximal bicliques. Discrete Applied Mathematics, 145:11–21, 2004. [2] J. Amilhastre, M.C. Vilarem, and P. Janssen. Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs. Discrete Applied Mathematics, 86:125–144, 1998. [3] P. Hammer. The conflict graph of a pseudo-boolean function. Technical report, Bell Labs, West Long Branch, NJ, 1978. [4] P. Hammer. Personal communication. Technical report, July 2006. [5] H. Müller. On edge perfectness and classes of bipartite graphs. Discrete Mathematics, 149:159–187, 1996. [6] J. Orlin. Contentment in graph theory: Covering graphs with cliques. Proceedings of the Koninklijke Nederlandse Akademie van Weteschappen, Series A, 80(5):406–424, 1977. 132 On the defensive k-alliance number of a graph Juan A. Rodrı́guez-Velázquez, a José M. Sigarreta b a Department of Computer Engineering and Mathematics Rovira i Virgili University of Tarragona Av. Paı̈sos Catalans 26, 43007 Tarragona, Spain b Departamento de Matemáticas Universidad Carlos III de Madrid Avda. de la Universidad 30, 28911 Legan (Madrid), Spain Key words: Alliances in graphs, Defensive alliances, Global alliances, Algebraic connectivity, Graph eigenvalues, Line graph 1 Introduction The mathematical properties of alliances in graphs were first studied by P. Kristiansen, S. M. Hedetniemi and S. T. Hedetniemi [5]. They proposed different types of alliances: namely, defensive alliances [3–5,10], offensive alliances [2,6–8] and dual alliances [6] or powerful alliances [1]. A generalization of these alliances called k-alliances was presented by K. H. Shafique and R. D. Dutton [11,12]. In this paper, we study the mathematical properties of defensive k-alliances. We begin by stating the terminology used. Throughout this article, Γ = (V, E) denotes a simple graph of order |V | = n and size |E| = m. We denote two adjacent vertices u and v by u ∼ v. For a nonempty set X ⊆ V , and a vertex v ∈ V , NX (v) denotes the set of neighbors v has in X: NX (v) := {u ∈ X : u ∼ v}, and the degree of v in X will be denoted by δX (v) = |NX (v)|. We denote the degree sequence of Γ by δ1 ≥ δ2 ≥ · · · ≥ δn . A nonempty set S ⊆ V is a defensive k-alliance in Γ = (V, E), k ∈ {−δ1 , . . . , δ1 }, if for every v ∈ S, δS (v) ≥ δS̄ (v) + k, where S̄ denotes the complement of the set S in V . A defensive (−1)-alliance is a defensive alliance and a defensive 0-alliance is a strong defensive alliance as defined in [5]. A defensive 0-alliance is also known as a cohesive set [13]. The defensive k-alliance number of Γ, denoted by ak (Γ), is defined as the minimum cardinality of a defensive k-alliance in Γ. Notice that ak+1 (Γ) ≥ ak (Γ). The defensive (−1)-alliance number of Γ is known as the alliance number of Γ and the defensive 0-alliance number is known as the 133 strong alliance number, [3–5]. Notice that if every vertex of Γ has even degree and k is odd, k = 2l − 1, then every defensive (2l − 1)-alliance in Γ is a defensive (2l)-alliance. Hence, in such a case, a2l−1 (Γ) = a2l (Γ). Analogously, if every vertex of Γ has odd degree and k is even, k = 2l, then every defensive (2l)-alliance in Γ is a defensive (2l + 1)-alliance. Hence, in such a case, a2l (Γ) = a2l+1 (Γ). For some graphs, there are some values of k > δn , such that defensive kalliances do not exist. For instance, for k ≥ 2 in the case of the star graph Sn , defensive k-alliances do not exist. Moreover, in any graph, there are defensive k-alliances for k ∈ {−δ1 , . . . , δn }. For instance, a defensive (δn )-alliance in Γ = (V, E) is V . In addition, if v ∈ V is a vertex of minimum degree, δ(v) = δn , then S = {v} is a defensive k-alliance for every k ≤ −δn . As ak (Γ) = 1 for k ≤ −δn , hereafter we only will consider the cases −δn ≤ k ≤ δ1 . Moreover, the results showed in this paper on ak (Γ), for δn ≤ k ≤ δ1 , are obtained by supposing that the graph Γ contains defensive k-alliances. 2 Results It was shown  in [5] that for any graph  Γ of order n and minimum degree δn , δn and a0 (Γ) ≤ n − δ2n . The following result generalizes a−1 (Γ) ≤ n − 2 the previous one to defensive k-alliances and provides lower bounds. Theorem 1 For every k ∈ {−δn , . . . , δ1 },    δn + k + 2 δn − k ≤ ak (Γ) ≤ n − . 2 2 We denote by Kn the complete graph of order n.   n+k+1 . Corollary 2 For every k ∈ {1 − n, . . . , n − 1}, ak (Kn ) = 2 Theorem 3 For every k, r ∈ Z such that −δn ≤ k ≤ δ1 and 0 ≤ r ≤ k+δn , 2 ak−2r (Γ) + r ≤ ak (Γ). n Corollary 4 Let t ∈ Z. If 1−δ ≤ t ≤ δ12−1 , then a2t−1 (Γ) + 1 ≤ a2t+1 (Γ) and, 2 2−δn δ1 if 2 ≤ t ≤ 2 , then a2(t−1) (Γ) + 1 ≤ a2t (Γ). Corollary 5 For every k ∈ {0, . . . , δn }, if k is even, then a−k (Γ) + k2 ≤ a0 (Γ) ≤ ak (Γ) − k2 and, if k is odd, then a−k (Γ) + k−1 ≤ a−1 (Γ) ≤ ak (Γ) − k+1 . 2 2 134 It was shown in [3,5] that for any graph Γ of order n, a−1 (Γ) ≤ a0 (Γ) ≤   n 2 + 1.    n 2 and  n+k+1 Theorem 6 For every k ∈ {−δn , . . . , 0}, ak (Γ) ≤ . 2 Notice that the above bound is attained, for instance, for the complete graph. The following theorem shows the relationship between the algebraic connectivity of a graph and its defensive k-alliance number. Theorem 7 For any connected graph Γ and for every k ∈ {−δn , . . . , δ1 },   n(µ + k + 1) ; ak (Γ) ≥ n+µ  ak (Γ) ≥   n(µ −  µ  δ1 −k 2   ) .   The above bounds are sharp as we can see in the following example. As the algebraic connectivity of the complete graph Γ = Kn is µ = n, the above  theorem gives the exact value of ak (Kn ) = n+k+1 . 2 Hereafter, we denote by L(Γ) = (Vl , El ) the line graph of Γ. Theorem 8 For any simple graph Γ of maximum degree δ1 , and for every  k k ∈ {2(1 − δ1 ), . . . , 0}, ak (L(Γ)) ≤ δ1 + 2 . One advantage of applying the above bound is that it requires only a little information about the graph Γ; just the maximum degree. The above bound is tight, as we will see  below, for any δ-regular graph, and k ∈ {2(1 − δ), . . . , 0}, ak (L(Γ)) = δ + k2 . Even so, we can improve this bound for the case of nonregular graphs. The drawback of such improvement is that we need to know more about Γ. Theorem 9 Let Γ be a simple graph, whose degree sequence is δ1 ≥ δ2 ≥ · · · ≥ δn . Let v ∈ V such that δ(v) = δ1 , let δv = max {δ(u)} and let δ∗ = u:u∼v min {δv }. For every k ∈ {2 − δ∗ − δ1 , . . . , δ1 − δ∗ }, ak (L(Γ)) ≤ v:δ(v)=δ1 Moreover, for every k ∈ {2 − δ1 − δ2 , . . . , δ1 + δ2 − 2},  δn +δn−1 +k 2   δ1 +δ∗ +k 2  . ≤ ak (L(Γ)). Corollary 10 For any δ-regular graph, δ > 0, and for every k ∈ {2(1 −   k δ), . . . , 0}, ak (L(Γ)) = δ + 2 . Corollary 11 For any (δ1 , δ2 )-semirregular bipartite  graph Γ, δ1 > δ2 , and for every k ∈ {2 − δ1 − δ2 , . . . , δ1 − δ2 }, ak (L(Γ)) = δ1 +δ22 +k . 135 The reader is referred to [10] for more details on a−1 (L(Γ)) and a0 (L(Γ)). Moreover, the proof of all the results included in this extended abstract can be found in [9]. References [1] R. Brigham; R. Dutton and S.Hedetniemi, A sharp lower bound on the powerful alliance number of Cm × Cn . Congr. Numer. 167 (2004) 57-63. [2] O. Favaron, G. Fricke, W. Goddard, S. Hedetniemi, S. T. Hedetniemi, P. Kristiansen, R. C. Laskar, R. D. Skaggs, Offensive alliances in graphs. Discuss. Math. Graph Theory 24 (2) (2004) 263-275. [3] G. H. Fricke, L. M. Lawson, T. W. Haynes, S. M. Hedetniemi and S. T. Hedetniemi, A Note on Defensive Alliances in Graphs. Bull. Inst. Combin. Appl. 38 (2003) 37-41. [4] T. W. Haynes, S. T. Hedetniemi and M. A. Henning, Global defensive alliances in graphs. Electron. J. Combin. 10 (2003) 139-146. [5] P. Kristiansen, S. M. Hedetniemi and S. T. Hedetniemi Alliances in graphs. J. Combin. Math. Combin. Comput. 48 (2004) 157-177. [6] J. A. Rodrı́guez and J. M. Sigarreta, Spectral study of alliances in graphs. Discuss. Math. Graph Theory. In press. [7] J. A. Rodrı́guez and J. M. Sigarreta, Offensive alliances in cubic graphs. International mathematical forum 1 (36) (2006) 1773-1782. [8] J. A. Rodrı́guez-Velázquez and J. M. Sigarreta, Global offensive alliances in graphs. Electronic Notes in Discrete Mathematics 25 (2006) 157-164. [9] J. A. Rodrı́guez-Velázquez, I. González-Yero and J. M. Sigarreta, Defensive k-alliance in graphs. http://arxiv.org/abs/math/0611180. [10] J. M. Sigarreta and J. A. Rodrı́guez, On defensive alliance and line graphs. Appied Mathematics Letters 19 (12) (2006) 1345-1350. [11] K. H. Shafique and R. D. Dutton, Maximum alliance-free and minimum alliancecover sets. Congr. Numer. 162 (2003) 139-146. [12] K. H. Shafique and R. Dutton, A tight bound on the cardinalities of maximun alliance-free and minimun alliance-cover sets. J. Combin. Math. Combin. Comput. 56 (2006) 139-145. [13] K. H. Shafique and R. D. Dutton, On satisfactory partitioning of graphs. Congress Numeratium 154 (2002) 183-194. 136 Relaxed Voting and Competitive Location on Trees under Monotonous Gain Functions J. Spoerhase H.–C. Wirth Lehrstuhl für Informatik I · Universität Würzburg · Am Hubland · 97074 Würzburg · Germany Key words: voting location, competitive location, trees We examine voting location problems on tree graphs: Facilities are to be placed into a graph in order to serve customers. The decision of placement is driven by an election process amongst the users, where the user preference is modeled by distances in the tree. In the problems under investigation we are particularly interested in stable solutions, i.e., where the chosen candidate is confronted with only weak oppositions. There are several measures for this stability, including Simpson and security score as defined later. It can be observed that in this model small differences between distances can have a huge impact on the result. This is avoided by making use of relaxed user preferences which allow for a tolerance in small distance differences [CM03]. The definitions in this section follow closely those of [CM03,CM02]. The input instance of the problem is given as a tree T = (V, E). A positive edge weight function d : E → R+ denotes lengths of edges and induces a distance function + d : V ×V → R+ 0 . Nonnegative node weights w : V → R0 specify the demand of individual user nodes. A nonnegative number α ∈ R+ 0 is used as a parameter to describe the users’ tolerance against small differences in distances: A user u prefers node x over node y, denoted by x ≺u y, if d(u, x) < d(u, y) − α. The user u is undecided, denoted by x ∼u y, if |d(u, x) − d(u, y)| ≤ α. The set of all users preferring y over x is U(y ≺ x) := { u | y ≺u x } and W (y ≺ x) :=  { w(u) | u ∈ U(y ≺ x) } denotes its weight. The problem family investigated here consists of placing two facilities, the candidate and the opposition, into the graph maximizing the individual influence. A gain function Φ : V × V → R maps a node pair (y, x) to the value Φ(y ≺ x) which measures in some sense the influence of an opposition node y after candidate node x has already been placed into the graph. The Φ-score Email addresses: spoerhase@informatik.uni-wuerzburg.de (J. Spoerhase), wirth@informatik.uni-wuerzburg.de (H.–C. Wirth). 137 of a candidate node x is defined as Φ(x) := maxy∈V Φ(y ≺ x). The Φ-score of a graph is defined as Φ∗ := minx∈V Φ(x). A Φ-solution of a graph is a candidate node x with Φ(x) = Φ∗ . This definition is a generalization of some existing problems including the Simpson and security problem on graphs (see Section 2). Single voting location on graphs under the unrelaxed user preference model, i.e. α = 0, and allowing placement of facilities on inner points of edges has been discussed in [HTW90]. For general graphs there are no fast algorithms known albeit the problems are polynomial time solvable. If the underlying graph is a tree, then most of these unrelaxed problems can be solved in linear time since they are equivalent to the median problem [HTW90]. This equivalence is no longer true if we turn over to the relaxed user preference model (α ≥ 0) which has been introduced in [CM03]. There is an algorithm for the relaxed Simpson problem on trees with running time O(n log n) as stated in [NSW07]. In this paper we provide fast algorithms for the relaxations of security solution, Stackelberg solution and Nash solution on trees. To this end we introduce the notion of monotonous gain function which allows to describe all those problems in a uniform way and to develop a general algorithmic framework. 1 The Algorithm A gain function Φ is called monotonous, if for all candidate-opposition pairs (x, y) and (x′ , y ′) where U(y ≺ x) ⊇ U(y ′ ≺ x′ ) and U(x ≺ y) ⊆ U(x′ ≺ y ′ ) we have that Φ(y ≺ x) ≥ Φ(y ′ ≺ x′ ). Observe that in particular the gain functions associated with Simpson and security problem are monotonous. We start with investigating the problem of computing the relaxed Φ-score of a node in a tree. Let x, y be nodes such that d(x, y) > α. We show that there are nodes x̃, ỹ (called front nodes) on path P (x, y) such that U(x ≺ y) = Tỹ (x̃) and U(y ≺ x) = Tx̃ (ỹ). This means that front nodes partition the tree into three subtrees which contain the users preferring the candidate, preferring the opposition, or being undecided. In all of our applications the value Φ(y ≺ x) only depends on the weight of such subtrees and hence can easily be accessed in O(1) after a global linear time preprocessing. Gain functions with this property are called front node computable. An α-neighbor of a candidate node x is any node y ∈ V at distance d(x, y) > α which is the first node on path P (x, y) with this property. The set of all αneighbors of x is denoted by N(x, α). We can show that the Φ-score of a node is attained essentially at α-neighbors, i.e., for each x there is a node y ∈ N(x, α) ∪ {x} such that Φ(x) = Φ(y ≺ x). 138 Then we give a complete characterization of the pair of front nodes associated with a given candidate-opposition pair and proceed with showing how all pairs of front nodes can be efficiently enumerated, using a depth first search traversal of the tree as a basis algorithm. Thus we state: For all front node computable monotonous gain functions Φ, the score Φ(x) of a node x can be computed in O(n log n). The Φ-score of the tree and a corresponding node x where this score is attained can be computed by a divide and conquer approach. The algorithm maintains a node xmin with currently minimal value Φ(xmin ) and a subtree Tactive which is guaranteed to contain all better nodes, if there exist any. In each iteration, the algorithm selects a pivot node x and restricts the further search space to a subtree hanging from a neighbor of the pivot node, called the guide node. We can show that if x is an arbitrary candidate node and v is a guide node of x, then all nodes x′ with Φ(x′ ) < Φ(x) must lie in subtree Tx (v) hanging from v. If we choose the pivot node as a median in the current active tree, this reduces the size of the active set by a constant factor in each iteration, which yields a logarithmic bound on the number of iterations: Theorem 1 A Φ-solution of a tree can be computed in time O(n(log n)2 ) for front node computable monotonous gain functions. Extension to Absolute Score and Solution The notions absolute Φ-score and absolute Φ-solution describe the problems similar to above where now candidate and opposition can be placed not only at nodes but also at inner points of edges [MF90] of the graph. We show how to modify the ideas outlined in the previous sections to derive an algorithm also for this edge placement situation. Theorem 2 An absolute Φ-solution of a tree can be computed in time O(n log n) for front node computable monotonous gain functions. 2 Applications Simpson If we select Φ(y ≺ x) := W (y ≺ x) we describe the Simpson problem. Applying the results from above yields an algorithm with running time O(n(log n)2 ). However we argue that in this special case the complete knowledge of the front nodes is not needed, which decreases the running time to O(n log n). Security By setting Φ(y ≺ x) := W (y ≺ x) − W (x ≺ y) for d(x, y) > α and Φ(y ≺ x) := −∞ otherwise one arrives at the security problem. (This distinguishes from the Simpson problem in that here undecided users have 139 no longer an influence on the quality of the solution.) Again, Φ satisfies all conditions so that Theorems 1 and 2 apply. Stackelberg In competitive location problems [ELT93] two providers, called leader and follower, sequentially locate facilities in a graph in order to maximize their market share. In the case of Stackelberg solutions, users prefer nearest facilities and split their demand equally amongst the two competing providers if they are undecided [HTW90]. It is easy to see that the problem of finding an optimal leader point is equivalent to computing the Φ-score of the tree where Φ(y ≺ x) := W (y ≺ x) + 21 W (y ∼ x). We can even generalize this problem to cover the case where undecided users split their demand between leader and following following an individual rule given by a function f : V → [0, 1]. Notice that in the case of f ≡ 0 and α = 0 we arrive at the (1|1)-centroid problem which is equivalent to the Simpson solution described earlier [Hak90]. Nash A pair (x, y) of nodes in a graph is called a Nash solution if no party can increase its payoff by moving to another location [HTW90]. The payoff is given as in the previous section as Φ(y ≺ x) = W (y ≺ x) + 12 W (y ∼ x). We can reduce the decicion problem whether a pair of nodes is in Nash equilibrium to evaluating the Φ-score at x and y. This results in algorithms with running time O(n log n) for the node placement and O(n) for the edge placement variations. References [CM02] C. M. Campos Rodríguez and J. A. Moreno Peréz · Multiple voting location problems · unpublished manuscript, 2002. [CM03] C. M. Campos Rodríguez and J. A. Moreno Peréz · Relaxation of the condorcet and simpson conditions in voting location · European Journal of Operations Research 145 (2003), 673–683. [ELT93] H. A. Eiselt, G. Laporte, and J.-F. Thisse · Competitive location models: A framework and bibliography · Transportation Science 27 (1993), no. 1, 44–54. [Hak90] S. L. Hakimi · Locations with spatial interactions: Competitive locations and games · in [MF90], 1990, pp. 439–478. [HTW90] P. Hansen, J.-F. Thisse, and R. E. Wendell · Equilibrium analysis for voting and competitive location problems · in [MF90], 1990, pp. 479–501. [MF90] P. B. Mirchandani and R. L. Francis · Discrete location theory · Series in Discrete Mathematics and Optimization, Wiley-Interscience, 1990. [NSW07] H. Noltemeier, J. Spoerhase, and H.-C. Wirth · Multiple voting location and single voting location on trees · European Journal of Operations Research 181 (2007), 654–667. 140