Abstract
In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth ≤k, (or equivalently, the class of partial k-trees), for fixed k. We introduce two classes of graph decision problems, LCC and ECC, and subclasses C-LCC, and C-ECC. We show that each problem in LCC (or C-LCC) is solvable in polynomial (O(n C)) time, when restricted to graphs with fixed upper bounds on the treewidth and degree; and that each problem in ECC (or C-ECC) is solvable in polynomial (O(n C)) time, when restricted to graphs with a fixed upper bound on the treewidth (with given corresponding tree-decomposition). Also, problems in C-LCC and C-ECC are solvable in polynomial time for graphs with a logarithmic treewidth, and in the case of C-LCC-problems, a fixed upper bound on the degree of the graph.
Also, we show for a large number of graph decision problems, their membership in LCC, ECC, C-LCC and/or C-ECC, thus showing the existence of O(n C) or polynomial algorithms for these problems, restricted to the graphs with bounded treewidth (and bounded degree). In several cases, C=1, hence our method gives in these cases linear algorithms.
For several NP-complete problems, and subclasses of the graphs with bounded treewidth, polynomial algorithms have been obtained. In a certain sense, the results in this paper unify these results.
Part of this work was carried out at the Dept. of Computer Science of the University of Utrecht, with financial support from the Foundation of Computer Science (S.I.O.N.) of the Netherlands Organization for the Advancement of Pure Research (Z.W.O.), and part of it was carried out at the Lab. of Computer Science of the Massachusetts Institute of Technology, with financial support of the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).
..!mcvax!ruuinf!hansb.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT, 25:2–23, 1985.
S. Arnborg, D. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth., 8:277–284, 1987.
S. Arnborg, J. Lagergren, and D. Seese. Which problems are easy for tree-decomposable graphs. 1987. Ext. abstract to appear in proc. ICALP 88.
S. Arnborg, J. Lagergren, and D. Seese. Which problems are easy for tree-decomposable graphs. 1988. In these proceedings.
S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems on graphs embedded in k-trees. TRITA-NA-8404, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 1984.
B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. In Proceedings 24th Ann. Symp. on Foundations of Computer Science, pages 265–273, IEEE Computer Society, Los Angeles, 1983, Preliminary version.
M. W. Bern, E. L. Lawler, and A. L. Wong. Why certain subgraph computations require only linear time. In Proc. 26th Symp. on Foundations of Computer Science, pages 117–125, 1985.
H. L. Bodlaender. Classes of Graphs with Bounded Treewidth. Technical Report RUU-CS-86-22, Dept. of Comp. Science, University of Utrecht, Utrecht, 1986.
H. L. Bodlaender. Dynamic programming algorithms on graphs with bounded tree-width. Tech. Rep., Lab. for Comp. Science, M.I.T., 1987.
H. L. Bodlaender. NC-algorithms for graphs with small treewidth. Technical Report RUU-CS-88-4, Dept. of Comp. Science, Univ. of Utrecht, Utrecht, 1988.
H. L. Bodlaender. Polynomial algorithms for Chromatic Index and Graph Isomorphism on partial k-trees. Tech. Rep. RUU-CS-87-17, Dept. of Comp. Sci., Univ. of Utrecht, 1987.
E. J. Cockayne, S. E. Goodman, and S. T. Hedetniemi. A linear algorithm for the domination number of a tree. Inform. Proc. Letters, 4:41–44, 1975.
C. J. Colbourn and L. K. Stewart. Dominating cycles in series-parallel graphs. Ans Combinatorica, 19A:107–112, 1985.
D. Coppersmith and U. Vishkin. Solving NP-hard problems in’ almost trees': vertex cover. Disc. Applied Match, 10:27–45, 1985.
G. Cornuéjols, D. Naddef, and W. R. Pulleyblank. Halin graphs and the traveling salesman problem. Math. Programming, 26:287–294, 1983.
M. R. Garey and D. S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.
Y. Gurevich, L. Stockmeyer, and U. Vishkin. Solving NP-hard problems on graphs that are almost trees and an application to facility location problems. J. Assoc. Comp. Mach., 31:459–473, 1984.
R. Hassin and A. Tamir. Efficient algorithms for optimization and selection on series-parallel graphs. SIAM J. Alg. Disc. Meth., 7:379–389, 1986.
S. T. Hedetniemi, R. Laskar, and J. Pfaff. A linear algorithm for the domination number of a cactus. Report 433, Dept. of Math. Sc., Clemson Univ., Clemson, S.C., 1983.
T. W. Hungerford. Algebra. Graduate Texts in Mathematics 73, Springer-Verlag, New York, 1974.
T. Kikuno, N. Yoshida, and Y. Kakuda. A linear algorithm for the domination number of a series-parallel graph. Discrete Appl. Math., 5:299–311, 1983.
R. Laskar, J. Pfaff, S. M. Hedetniemi, and S. T. Hedetniemi. On the algorithmic complexity of total domination. SIAM J. Alg. Disc. Meth., 5:420–425, 1984.
B. Monien and I. Sudborough. Min cut is NP-complete for edge weighted trees. In Proc. of Int. Conf. Automata, Languages, and Programming ICALP '86, pages 265–274, Springer Verlag Lecture Notes in Comp. Science, Vol 226, 1986.
B. Monien and I. H. Sudborough. Bandwidth-constrained NP-complete problems. In Proc. 13th Ann. ACM Symp. on Theory of Computing, pages 207–217, Assoc. For Computing Machinery, New York, 1981.
A. Proskurowski and M. M. Sysło. Efficient vertex-and edge-coloring of outerplanar graphs. Report UO-CIS-TR-82-5, Dept. of Computer and Information Sc., Univ. of Oregon Eugene, Ore., 1982.
N. Robertson and P. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. of Algorithms, 7:309–322, 1986.
N. Robertson and P. Seymour. Graph minors. X. Obstructions to tree-decompositions. 1986. Manuscript.
N. Robertson and P. Seymour. Graph minors. XIII. The disjoint paths problem. 1986. Manuscript.
P. Scheffler and D. Seese. A combinatorial and logical approach to linear-time computability. 1986. Extended abstract.
I. H. Sudborough. “Cutwidth” and related graph problems. Bulletin of the EATCS, 79–110, Feb. 1987.
M. M. Sysło. NP-complete problems on some tree-structured graphs: a review. In M. Nagl and J. Perl, editors, Proc. WG'83 International Workshop on Graph Theoretic Concepts in Computer Science, pages 342–353, Univ. Verlag Rudolf Trauner, Linz, West Germany, 1983.
M. M. Sysło. The subgraph isomorphism problem for outerplanar graphs. Theor. Comput. Science, 17:91–97, 1982.
K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM, 29:623–641, 1982.
J. Wald and C. Colbourn. Steiner trees, partial 2-trees, and minimum IFI networks. Networks, 13:159–167, 1983.
J. Wald and C. J. Colbourn. Steiner trees in outerplanar graphs. In Proc. 13th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, Winnipeg, Ont., 1982.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodlaender, H.L. (1988). Dynamic programming on graphs with bounded treewidth. In: Lepistö, T., Salomaa, A. (eds) Automata, Languages and Programming. ICALP 1988. Lecture Notes in Computer Science, vol 317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19488-6_110
Download citation
DOI: https://doi.org/10.1007/3-540-19488-6_110
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-19488-0
Online ISBN: 978-3-540-39291-0
eBook Packages: Springer Book Archive