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.
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.
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
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.
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.
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.
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
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.
Uploads
Papers by Rainer Burkard