Numerische Methoden bei Optimierungsaufgaben Band 3, 1977
Ein algebraischer Ansatz fur die Zielfunktionen von Transportproblemen und Flusproblemen in Netzw... more Ein algebraischer Ansatz fur die Zielfunktionen von Transportproblemen und Flusproblemen in Netzwerken fuhrt auf ein Problem, das die klassischen Falle von Summenzielfunktion und Engpaszielfunktion enthalt. Zur Losung des allgemeinen Transportproblemes konnen “zulassige Transformationen” herangezogen werden, wahrend das allgemeine Flusproblem durch verallgemeinerte Flusalgorithmen gelost werden kann. Der algebraische Ansatz gewahrt nicht nur Einblick in die Struktur der Probleme sondern erklart auch ihr verschiedenes numerisches Verhalten.
Mathematische Operationsforschung und Statistik. Series Optimization, 1984
... For \ 7 < 7 , nyly efficl;& soisttio:z z* ?,Cit,G ye ~ ~ pe ~ t to (q , z,; a?... more ... For \ 7 < 7 , nyly efficl;& soisttio:z z* ?,Cit,G ye ~ ~ pe ~ t to (q , z,; a?;fi ? fi ~ gtl p zp, xox2qc/-"ltiz.e I "' ih7;;7-u,jt:;t(C -7-3 i7-- LIAllue, these is u .iizc?nEjer .is, G-;?,o= 1) s.iir;]L that zk 2.i~ opii.mcil suiuiio7z to . ,: min I(V; Z) = min ? ...
The 1-median problem on a network asks for a vertex minimizing the sum of the weighted shortest p... more The 1-median problem on a network asks for a vertex minimizing the sum of the weighted shortest path distances from itself to all other vertices, each associated with a certain positive weight. We allow fornegative weights as well and devise an exact algorithm for the resulting ‘pos/neg-weighted’ problem defined on a cactus. The algorithm visits every vertex just once and runs thus in linear time.
An efficiant or n~~dumi~ated solution to a multicriteria minimization problem is a feasible solut... more An efficiant or n~~dumi~ated solution to a multicriteria minimization problem is a feasible solution for which a decrease in the value of any criterion can only be obtained if the value of at least one other criterion is increased. Interest in the conditions under which solutions to multicriteria programming problems can be shown to be efficient has grown enormously in recent years, not least of all due to the awareness that significant planning problems can only be meaningfully modelled if multiple measures of effectiveness are considered. However, the broad class of multicriteria 8-l programming problems. which are of great practical importance, has received only limited attention in the literature. We establish the identity of the set of efficient solutions with respect to such multicriteria programming problems with any criteriu and any construint set and the set of optimal solutions to a parametrized unicriterion problem incorporating these criteria. Illustrative numerical examples are provided.
Burkard, R.E. and W. Sandholzer, Efficiently solvable special cases of bottleneck travelling sale... more Burkard, R.E. and W. Sandholzer, Efficiently solvable special cases of bottleneck travelling salesman problems, Discrete Applied Mathematics 32 (1991) 61-76. The paper investigates bottleneck travelling salesman problems (BTSP) which can be solved in polynomial time. At first a BTSP whose cost matrix is a circulant is treated. It is shown that in the symmetric case such a BTSP can be solved in O(n log n) time. Secondly conditions are derived which guarantee that an optimal solution is a pyramidal tour. Thus this problem can be solved in O(n2) time. Finally it is shown that a BTSP with cost matrix C=(c;,), where cl1 =a, b, with a, 5 ... <a, and b,z ... 26, can be solved in O(n*) time.
For assignment problems a class of objective functions is studied by algebraic methods and charac... more For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.
This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It di... more This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior. Moreover, it also considers problems related to the QAP, e.g. the biquadratic assignment problem, and discusses the relationship between the QAP and other well known combinatorial optimization problems, e.g. the traveling salesman problem, the graph partitioning problem, etc. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.
The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment proble... more The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). As for any hard optimization problem also for BiQAP, a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which, possibly, yield a good trade off between the solution quality and the time and memory requirements. In this paper we describe several heuristics for BiQAPs, in particular pair exchange algorithms (improvement methods) and variants of simulated annealing and taboo search. We implement these heuristics as C codes and analyze their performances.
... A general solution method for algebraic linear assignment problems; 4.1.1. Algorithm for solv... more ... A general solution method for algebraic linear assignment problems; 4.1.1. Algorithm for solving linear algebraic assignment problems. 4.2. Linear sum assignment problems; 4.3. ... 4.1. A general solution method for algebraic linear assignment problems. ...
... for quadratic Boolean programs with applications to quadratic assignment problems Rainer E. B... more ... for quadratic Boolean programs with applications to quadratic assignment problems Rainer E. BURKARD Institut fur Mathematik, Technische Universitdt Graz, A8010 Graz, Austria Tilman BONNIGER Rechenzentrum der ... Therefore there is a great demand for good heuristics. ...
This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We ar... more This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set E, a class F of nonempty subsets of E, a weight w:E→R+ and a cost c:E→R+. For each e∈E, c(e) stands for the cost of reducing weight w(e) by one unit. For each subset F∈F, the bottleneck weight of F is
An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j ... more An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j < s d n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
Consider a network \(\mathcal{N}\) =(G, c, τ) whereG=(N, A) is a directed graph andc ij andτ ij ,... more Consider a network \(\mathcal{N}\) =(G, c, τ) whereG=(N, A) is a directed graph andc ij andτ ij , respectively, denote the capacity and the transmission time of arc (i, j) ∈A. The quickest flow problem is then to determine for a given valueυ the minimum numberT(υ) of time units that are necessary to transmit (send)υ units of flow in \(\mathcal{N}\) from a given sources to a given sinks′. In this paper we show that the quickest flow problem is closely related to the maximum dynamic flow problem and to linear fractional programming problems. Based on these relationships we develop several polynomial algorithms and a strongly polynomial algorithm for the quickest flow problem. Finally we report computational results on the practical behaviour of our metholds. It turns out that some of them are practically very efficient and well-suited for solving large problem instances.
ABSTRACT Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment... more ABSTRACT Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment problem (BQAP) which generalizes the wellknown quadratic assignment problem (QAP). The BQAP is to minimize a weighted sum of products of four variables subject to assignment constraints on the variables. We give two integer programming formulations for the problem and design lower bounds for the optimal solution value. These lower bounds are tested computationally on BQAP instances with known objective function value. Finally the asymptotic behaviour of BQAPs is analyzed. It turns out that the ratio between the best and the worst objective function values tends in probability to one when the size of the problem tends to infinity.
The inverse p-median problem consists in changing the weights of the customers of a p-median loca... more The inverse p-median problem consists in changing the weights of the customers of a p-median location problem at minimum cost such that a set of p prespeciÿed suppliers becomes the p-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that the discrete version of an inverse p-median problem can be formulated as a linear program. Therefore, it is polynomially solvable for ÿxed p even in the case of mixed positive and negative customer weights. In the case of trees with nonnegative vertex weights, the inverse 1-median problem is solvable in a greedy-like fashion. In the plane, the inverse 1-median problem can be solved in O(n log n) time, provided the distances are measured in l1or l∞-norm, but this is not any more true in R 3 endowed with the Manhattan metric.
Let the graph G = (V, E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive ... more Let the graph G = (V, E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost such that a prespecified vertex becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that this problem can be formulated as a linear program with bounded variables and a special structure of the constraint matrix: the columns of the linear program can be partitioned into two classes in which they are monotonically decreasing. This allows to solve the problem in O(n 2 )-time.
This paper deals with facility location problems with pos=neg weights in trees. We consider two d... more This paper deals with facility location problems with pos=neg weights in trees. We consider two di erent objective functions which model two di erent ways to handle obnoxious facilities. If we minimize the overall sum of the minimum weighted distances of the vertices from the facilities, the optimal solution has nice combinatorial properties, e.g., vertex optimality. For the pos=neg 2-median problem on a network with n vertices, these properties can be exploited to derive an O(n 2 ) algorithm for trees, an O(n log n) algorithm for stars and a linear algorithm for paths. For the p-median problem with pos=neg weights on a path we give an O(pn 2 ) algorithm. If we minimize the overall sum of the weighted minimum distances of the vertices from the facilities, we can show that there exists a ÿnite set of O(n 3 ) points in the tree which contains the locations of facilities in an optimal solution. This leads to an O(n 3 ) algorithm for ÿnding the optimum 2-medians in a tree. The complexity can be reduced to O(n 2 ), if the medians are restricted to vertices or if the tree is a path. ?
Numerische Methoden bei Optimierungsaufgaben Band 3, 1977
Ein algebraischer Ansatz fur die Zielfunktionen von Transportproblemen und Flusproblemen in Netzw... more Ein algebraischer Ansatz fur die Zielfunktionen von Transportproblemen und Flusproblemen in Netzwerken fuhrt auf ein Problem, das die klassischen Falle von Summenzielfunktion und Engpaszielfunktion enthalt. Zur Losung des allgemeinen Transportproblemes konnen “zulassige Transformationen” herangezogen werden, wahrend das allgemeine Flusproblem durch verallgemeinerte Flusalgorithmen gelost werden kann. Der algebraische Ansatz gewahrt nicht nur Einblick in die Struktur der Probleme sondern erklart auch ihr verschiedenes numerisches Verhalten.
Mathematische Operationsforschung und Statistik. Series Optimization, 1984
... For \ 7 &lt; 7 , nyly efficl;&amp; soisttio:z z* ?,Cit,G ye ~ ~ pe ~ t to (q , z,; a?... more ... For \ 7 &lt; 7 , nyly efficl;&amp; soisttio:z z* ?,Cit,G ye ~ ~ pe ~ t to (q , z,; a?;fi ? fi ~ gtl p zp, xox2qc/-&quot;ltiz.e I &quot;&#x27; ih7;;7-u,jt:;t(C -7-3 i7-- LIAllue, these is u .iizc?nEjer .is, G-;?,o= 1) s.iir;]L that zk 2.i~ opii.mcil suiuiio7z to . ,: min I(V; Z) = min ? ...
The 1-median problem on a network asks for a vertex minimizing the sum of the weighted shortest p... more The 1-median problem on a network asks for a vertex minimizing the sum of the weighted shortest path distances from itself to all other vertices, each associated with a certain positive weight. We allow fornegative weights as well and devise an exact algorithm for the resulting ‘pos/neg-weighted’ problem defined on a cactus. The algorithm visits every vertex just once and runs thus in linear time.
An efficiant or n~~dumi~ated solution to a multicriteria minimization problem is a feasible solut... more An efficiant or n~~dumi~ated solution to a multicriteria minimization problem is a feasible solution for which a decrease in the value of any criterion can only be obtained if the value of at least one other criterion is increased. Interest in the conditions under which solutions to multicriteria programming problems can be shown to be efficient has grown enormously in recent years, not least of all due to the awareness that significant planning problems can only be meaningfully modelled if multiple measures of effectiveness are considered. However, the broad class of multicriteria 8-l programming problems. which are of great practical importance, has received only limited attention in the literature. We establish the identity of the set of efficient solutions with respect to such multicriteria programming problems with any criteriu and any construint set and the set of optimal solutions to a parametrized unicriterion problem incorporating these criteria. Illustrative numerical examples are provided.
Burkard, R.E. and W. Sandholzer, Efficiently solvable special cases of bottleneck travelling sale... more Burkard, R.E. and W. Sandholzer, Efficiently solvable special cases of bottleneck travelling salesman problems, Discrete Applied Mathematics 32 (1991) 61-76. The paper investigates bottleneck travelling salesman problems (BTSP) which can be solved in polynomial time. At first a BTSP whose cost matrix is a circulant is treated. It is shown that in the symmetric case such a BTSP can be solved in O(n log n) time. Secondly conditions are derived which guarantee that an optimal solution is a pyramidal tour. Thus this problem can be solved in O(n2) time. Finally it is shown that a BTSP with cost matrix C=(c;,), where cl1 =a, b, with a, 5 ... <a, and b,z ... 26, can be solved in O(n*) time.
For assignment problems a class of objective functions is studied by algebraic methods and charac... more For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.
This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It di... more This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior. Moreover, it also considers problems related to the QAP, e.g. the biquadratic assignment problem, and discusses the relationship between the QAP and other well known combinatorial optimization problems, e.g. the traveling salesman problem, the graph partitioning problem, etc. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.
The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment proble... more The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). As for any hard optimization problem also for BiQAP, a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which, possibly, yield a good trade off between the solution quality and the time and memory requirements. In this paper we describe several heuristics for BiQAPs, in particular pair exchange algorithms (improvement methods) and variants of simulated annealing and taboo search. We implement these heuristics as C codes and analyze their performances.
... A general solution method for algebraic linear assignment problems; 4.1.1. Algorithm for solv... more ... A general solution method for algebraic linear assignment problems; 4.1.1. Algorithm for solving linear algebraic assignment problems. 4.2. Linear sum assignment problems; 4.3. ... 4.1. A general solution method for algebraic linear assignment problems. ...
... for quadratic Boolean programs with applications to quadratic assignment problems Rainer E. B... more ... for quadratic Boolean programs with applications to quadratic assignment problems Rainer E. BURKARD Institut fur Mathematik, Technische Universitdt Graz, A8010 Graz, Austria Tilman BONNIGER Rechenzentrum der ... Therefore there is a great demand for good heuristics. ...
This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We ar... more This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set E, a class F of nonempty subsets of E, a weight w:E→R+ and a cost c:E→R+. For each e∈E, c(e) stands for the cost of reducing weight w(e) by one unit. For each subset F∈F, the bottleneck weight of F is
An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j ... more An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j < s d n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
Consider a network \(\mathcal{N}\) =(G, c, τ) whereG=(N, A) is a directed graph andc ij andτ ij ,... more Consider a network \(\mathcal{N}\) =(G, c, τ) whereG=(N, A) is a directed graph andc ij andτ ij , respectively, denote the capacity and the transmission time of arc (i, j) ∈A. The quickest flow problem is then to determine for a given valueυ the minimum numberT(υ) of time units that are necessary to transmit (send)υ units of flow in \(\mathcal{N}\) from a given sources to a given sinks′. In this paper we show that the quickest flow problem is closely related to the maximum dynamic flow problem and to linear fractional programming problems. Based on these relationships we develop several polynomial algorithms and a strongly polynomial algorithm for the quickest flow problem. Finally we report computational results on the practical behaviour of our metholds. It turns out that some of them are practically very efficient and well-suited for solving large problem instances.
ABSTRACT Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment... more ABSTRACT Motivated by a problem arising in VLSI synthesis we introduce the biquadratic assignment problem (BQAP) which generalizes the wellknown quadratic assignment problem (QAP). The BQAP is to minimize a weighted sum of products of four variables subject to assignment constraints on the variables. We give two integer programming formulations for the problem and design lower bounds for the optimal solution value. These lower bounds are tested computationally on BQAP instances with known objective function value. Finally the asymptotic behaviour of BQAPs is analyzed. It turns out that the ratio between the best and the worst objective function values tends in probability to one when the size of the problem tends to infinity.
The inverse p-median problem consists in changing the weights of the customers of a p-median loca... more The inverse p-median problem consists in changing the weights of the customers of a p-median location problem at minimum cost such that a set of p prespeciÿed suppliers becomes the p-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that the discrete version of an inverse p-median problem can be formulated as a linear program. Therefore, it is polynomially solvable for ÿxed p even in the case of mixed positive and negative customer weights. In the case of trees with nonnegative vertex weights, the inverse 1-median problem is solvable in a greedy-like fashion. In the plane, the inverse 1-median problem can be solved in O(n log n) time, provided the distances are measured in l1or l∞-norm, but this is not any more true in R 3 endowed with the Manhattan metric.
Let the graph G = (V, E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive ... more Let the graph G = (V, E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost such that a prespecified vertex becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that this problem can be formulated as a linear program with bounded variables and a special structure of the constraint matrix: the columns of the linear program can be partitioned into two classes in which they are monotonically decreasing. This allows to solve the problem in O(n 2 )-time.
This paper deals with facility location problems with pos=neg weights in trees. We consider two d... more This paper deals with facility location problems with pos=neg weights in trees. We consider two di erent objective functions which model two di erent ways to handle obnoxious facilities. If we minimize the overall sum of the minimum weighted distances of the vertices from the facilities, the optimal solution has nice combinatorial properties, e.g., vertex optimality. For the pos=neg 2-median problem on a network with n vertices, these properties can be exploited to derive an O(n 2 ) algorithm for trees, an O(n log n) algorithm for stars and a linear algorithm for paths. For the p-median problem with pos=neg weights on a path we give an O(pn 2 ) algorithm. If we minimize the overall sum of the weighted minimum distances of the vertices from the facilities, we can show that there exists a ÿnite set of O(n 3 ) points in the tree which contains the locations of facilities in an optimal solution. This leads to an O(n 3 ) algorithm for ÿnding the optimum 2-medians in a tree. The complexity can be reduced to O(n 2 ), if the medians are restricted to vertices or if the tree is a path. ?
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