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Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms 1

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9th international conference on integer programming and combinatorial optimization, ipco 2002

作者： Richard, Jean-Philippe P. De Farias Jr., Ismael R. Nemhauser, George L. School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta GA 30332-0205 United States Center for Operations Research and Econometrics 34 Voie du Roman Pays 1348 Louvain-La-Neuve Belgium

ISBN: (数字)9783540478676

ISBN: (纸本)9783540478676

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables. We introduce the concept of superlinear inequalities and show that our lifting scheme can be significantly simplified for them. Finally, we show that superlinearity results can be generalized to nonsuperlinear inequalities when the coefficients of the continuous variables lifted are large. © 2002 Springer-Verlag Berlin Heidelberg.

关键词： integer programming

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An approximation scheme for the two-stage, two-dimensional bin packing problem

An approximation scheme for the two-stage, two-dimensional b...

引用

9th international integer programming and combinatorial optimization conference, ipco 2002

作者： Caprara, Alberto Lodi, Andrea Monaci, Michele D.E.I.S. Università di Bologna Viale Risorgimento 2 Bologna40136 Italy

ISBN: (纸本)9783540478676

We present an asymptotic PTAS for Two-Dimensional Bin Packing, which requires packing (or cutting) a given set of rectangles from the minimum number of square bins, with the further restriction that cutting the rectangles from the bins can be done in two stages, as is frequently the case in real-world applications. To the best of our knowledge, this is the first approximation scheme for a nontrivial twodimensional (and real-world) generalization of classical one-dimensional Bin Packing in which rectangles have to be packed in (finite) squares. A simplification of our method yields an asymptotic PTAS for the two-stage packing of rectangles in a bin of unit width and infinite height. Moreover, we point out that our method may lead to a better approximation guarantee for Two-Dimensional Bin Packing without stage restrictions, provided some structural property holds. © Springer-Verlag Berlin Heidelberg 2002.

关键词： integer programming

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Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

Lifted inequalities for 0-1 mixed integer programming: Basic...

引用

9th international integer programming and combinatorial optimization conference, ipco 2002

作者： Richard, Jean-Philippe P. de Farias, Ismael R. Nemhauser, George L. School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332-0205 United States Center for Operations Research and Econometrics 34 Voie du Roman Pays Louvain-La-Neuve1348 Belgium School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332-0205 United States

ISBN: (纸本)9783540478676

关键词： integer programming

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A polyhedral approach to surface reconstruction from planar contours 1

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9th international conference on integer programming and combinatorial optimization, ipco 2002

作者： Althaus, Ernst Fink, Christian International Computer Science Institute 1947 Center St. Berkeley CA 94704-1198 United States Max-Planck-Institute für Informatik Im Stadtwald 66123 Saarbrücken Germany

ISBN: (数字)9783540478676

ISBN: (纸本)9783540478676

We investigate the problem of reconstruction a surface given its contours on parallel slices. We present a branch-and-cut algorithm which computes the surface with the minimal area. this surface is assumed to be the best reconstruction since a long time. Nevertheless there were no algorithms to compute this surface. Our experiments show that the running time of our algorithm is very reasonable and that the computed surfaces are highly similar to the original surfaces. © 2002 Springer-Verlag Berlin Heidelberg.

关键词： Surface reconstruction

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On preemptive resource constrained scheduling: Polynomial-time approximation schemes

On preemptive resource constrained scheduling: Polynomial-ti...

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9th international integer programming and combinatorial optimization conference, ipco 2002

作者： Jansen, Klaus Porkolab, Lorant Institut für Informatik und praktische Mathematik Christian-Albrechts-Universität zu Kiel Kiel24 098 Germany PricewaterhouseCoopers London United Kingdom

ISBN: (纸本)9783540478676

We study resource constrained scheduling problems where the objective is to compute feasible preemptive schedules minimizing the makespan and using no more resources than what are available. We present approximation schemes along with some inapproximibility results showing how the approximability of the problem changes in terms of the number of resources. the results are based on linear programming formulations (though with exponentially many variables) and some interesting connections between resource constrained scheduling and (multidimensional, multiple-choice, and cardinality constrained) variants of the classical knapsack problem. In order to prove the results we generalize a method by Grigoriadis et al. for the max-min resource sharing problem to the case with weak approximate block solvers (i.e. with only constant, logarithmic, or even worse approximation ratios). Finally we present applications of the above results in fractional graph coloring and multiprocessor task scheduling. © Springer-Verlag Berlin Heidelberg 2002.

关键词： Linear programming

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A polyhedral study of the cardinality constrained knapsack problem

A polyhedral study of the cardinality constrained knapsack p...

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9th international conference on integer programming and combinatorial optimization, ipco 2002

作者： De Farias Jr., Ismael R. Nemhauser, George L. CORE 34 Voie du Roman Pays 1348 Louvain-la-Neuve Belgium School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta GA 30332 United States

ISBN: (纸本)9783540478676

A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. this structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. © 2002 Springer-Verlag Berlin Heidelberg.

关键词： combinatorial optimization

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the semidefinite relaxation of the κ-partition polytope is strong

The semidefinite relaxation of the κ-partition polytope is ...

引用

9th international conference on integer programming and combinatorial optimization, ipco 2002

作者： Eisenblätter, Andreas Takustr. 7 D-14195 Berlin Germany

ISBN: (纸本)9783540478676

Radio frequency bandwidth has become a very scarce resource. this holds true in particular for the popular mobile communication system GSM. Carefully planning the use of the available frequencies is thus of great importance to GSM network operators. Heuristic optimization methods for this task are known, which produce frequency plans causing only moderate amounts of disturbing interference in many typical situations. In order to thoroughly assess the quality of the plans, however, lower bounds on the unavoidable interference are in demand. the results obtained so far using linear programming and graph theoretic arguments do not suffice. By far the best lower bounds are currently obtained from semidefinite programming. the link between semidefinite programming and the bound on unavoidable interference in frequency planning is the semidefinite relaxation of the graph minimum κ-partition problem. Here, we take first steps to explain the surprising strength of the semidefinite relaxation. this bases on a study of the solution set of the semidefinite relaxation in relation to the circumscribed κ-partition polytope. Our focus is on the huge class of hypermetric inequalities, which are valid and in many cases facet-defining for the κ-partition polytope. We show that a "slightly shifted version" of the hypermetric inequalities is implicit to the semidefinite relaxation. In particular, no feasible point for the semidefinite relaxation violates any of the facet-defining triangle inequalities for the κ-partition polytope by more than √2 - 1 or any of the (exponentially many) facet-defining clique constraints by 1/2 or more. © 2002 Springer-Verlag Berlin Heidelberg.

关键词： Linear programming

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A polyhedral study of the cardinality constrained knapsack problem

A polyhedral study of the cardinality constrained knapsack p...

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9th international integer programming and combinatorial optimization conference, ipco 2002

作者： de Farias, Ismael R. Nemhauser, George L. CORE 34 Voie du Roman Pays Louvain-la-Neuve1348 Belgium School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332 United States

ISBN: (纸本)9783540478676

关键词： combinatorial optimization

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the semidefinite relaxation of the k-partition polytope is strong

The semidefinite relaxation of the k-partition polytope is s...

引用

9th international integer programming and combinatorial optimization conference, ipco 2002

作者： Eisenblätter, Andreas Takustr. 7 BerlinD-14195 Germany

ISBN: (纸本)9783540478676

Radio frequency bandwidth has become a very scarce resource. this holds true in particular for the popular mobile communication system GSM. Carefully planning the use of the available frequencies is thus of great importance to GSM network operators. Heuristic optimization methods for this task are known, which produce frequency plans causing only moderate amounts of disturbing interference in many typical situations. In order to thoroughly assess the quality of the plans, however, lower bounds on the unavoidable interference are in demand. the results obtained so far using linear programming and graph theoretic arguments do not suffice. By far the best lower bounds are currently obtained from semidefinite programming. the link between semidefinite programming and the bound on unavoidable interference in frequency planning is the semidefinite relaxation of the graph minimum k-partition problem. Here, we take first steps to explain the surprising strength of the semidefinite relaxation. this bases on a study of the solution set of the semidefinite relaxation in relation to the circumscribed k-partition polytope. Our focus is on the huge class of hypermetric inequalities, which are valid and in many cases facet-defining for the k-partition polytope. We show that a "slightly shifted version" of the hypermetric inequalities is implicit to the semidefinite relaxation. In particular, no feasible point for the semidefinite relaxation violates any of the facet-defining triangle inequalities for the k-partition polytope by more than √2 - 1 or any of the (exponentially many) facet-defining clique constraints by 1/2 or more. © Springer-Verlag Berlin Heidelberg 2002.

关键词： Linear programming

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integer programming and combinatorial optimization

引用

丛书名： Lecture notes in computer science

2001年

作者： Karen Aardal Bert Gerards (eds.).

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