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Generating cuts from multiple-term disjunctions

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8th international integer programming and combinatorial optimization conference, ipco 2001

作者： Perregaard, Michael Balas, Egon Carnegie Mellon University PittsburghPA United States

ISBN: (纸本)3540422250

the traditional approach towards generating lift-and-project cuts involves solving a cut generating linear program (CGLP) that grows prohibitively large if a multiple-term disjunction is used instead of the classical dichotomy. In this paper we present an alternative approach towards generating lift-and-project cuts that solves the cut generation problem in the original space (i.e. does not introduce any extra variables). We present computational results that show our method to be superior to the classical approach. As soon as the number of terms in the disjunction grows larger than 4, the benefit of not solving the large CGLP directly is clear, and the gap grows larger as more terms are included in the disjunction. © Springer-Verlag Berlin Heidelberg 2001.

关键词： Linear programming

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integer programming and combinatorial optimization: 7th international ipco conference Graz, Austria, June 9-11, 1999 Proceedings 7th

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Cornuéjols, Gérard Burkard, Rainer E. Woeginger, Gerhard J. GSIA Carnegie Mellon University Schenley Park PittsburghPA United States Institut für Mathematik Technische Universität Graz Steyrergasse 30 Graz Austria

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6th international conference on integer programming and combinatorial optimization, ipco 1998

6th International Conference on Integer Programming and Comb...

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6th international integer programming and combinatorial optimization conference, ipco 1998

ISBN: (纸本)354064590X

the proceedings contain 32 papers. the special focus in this conference is on Edge Connectivity and Algorithms. the topics include: A characterization of weakly bipartite graphs;bipartite designs;characterizing noninteger polyhedra with 0-1 constraints;a theorem of truemper;the generalized stable set problem for claw-free bidirected graphs;on a min-max theorem of cacti;edge-splitting and edge-connectivity augmentation in planar graphs;a new bound for the 2-edge connected subgraph problem;an approximation algorithm for 2-edge connected spanning subgraphs;multicuts in unweighted graphs with bounded degree and bounded tree-width;approximating disjoint-path problems using greedy algorithms and packing integer programs;approximation algorithms for the mixed postman problem;improved approximation algorithms for uncapacitated facility location;the maximum traveling salesman problem under polyhedral norms;polyhedral combinatorics of benzenoid problems;consecutive ones and a betweenness problem in computational biology;solving a linear diophantine equation with lower and upper bounds on the variables;the intersection of knapsack polyhedra and extensions;new classes of lower bounds for bin packing problems;solving integer and disjunctive programs by lift and project;a class of hard small 0-1 programs;building chain and cactus representations of all minimum cuts from hao-orlin in the same asymptotic run time;simple generalized maximum flow algorithms;the pseudoflow algorithm and the pseudoflow-based simplex for the maximum flow problem;an implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow and non-approximability results for scheduling problems with minsum criteria.

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On the number of iterations for Dantzig-Wolfe optimization and packing-covering approximation algorithms 7th

引用

7th international conference on integer programming and combinatorial optimization (ipco VII)

作者： Klein, P Young, N Brown Univ Providence RI 02912 USA Dartmouth Coll Hanover NH 03755 USA

ISBN: (纸本)3540660194

We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A∈?m×n, b∈?m and a polytope P $ \subseteq$ \subseteq ? n , the fractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ∈-approximate solution to this problem is an x ∈ P such that Ax ≤ (1+∈)b. An ∈-relaxed decision procedure always finds an ∈-approximate solution if an exact solution exists.

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Optimizing over all combinatorial embeddings of a planar graph 7th

引用

7th international conference on integer programming and combinatorial optimization (ipco VII)

作者： Mutzel, P Weiskircher, R Max Planck Inst Informat Saarbrucken Germany

ISBN: (纸本)3540660194

We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. the motivation for studying this problem arises in graph drawing, where the chosen embedding has an important influence on the aesthetics of the drawing. We characterize the set of all possible embeddings of a given biconnected planar graph G by means of a system of linear inequalities with {0, 1}-variables corresponding to the set of those cycles in G which can appear in a combinatorial embedding. this system of linear inequalities can be constructed recursively using SPQR-trees and a new splitting operation. Our computational results on two benchmark sets of graphs are surprising: the number of variables and constraints seems to grow only linearly with the size of the graphs although the number of embeddings grows exponentially. For all tested graphs (up to 500 vertices) and linear objective functions, the resulting integer linear programs could be generated within 10 minutes and solved within two seconds on a Sun Enterprise 10000 using CPLEX.

关键词： integer programming

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An introduction to empty lattice simplices 7th

引用

7th international conference on integer programming and combinatorial optimization (ipco VII)

作者： Sebo, A CNRS Lab Leibniz IMAG Grenoble France Kyoto Univ Math Sci Res Inst Kyoto 60601 Japan

ISBN: (纸本)3540660194

We study simplices whose vertices lie on a lattice and have no other lattice points. Such 'empty lattice simplices' come up in the theory of integer programming, and in some combinatorial problems. they have been investigated in various contexts and under varying terminology by Reeve, White, Scarf, Kannan and Lovasz, Reznick, Kantor, Naase and Ziegler, etc. Can the 'emptiness' of lattice simplices be 'well-characterized' ? Is their 'lattice-width' small ? Do the integer points of the parallelepiped they generate have a particular structure ? the 'good characterization' of empty lattice simplices occurs to be open in general ! We provide a polynomial algorithm for deciding when a given integer 'knapsack' or 'partition' lattice simplex is empty. More generally, we ask for a characterization of linear inequalities satisfied by the lattice points of a lattice parallelepiped. We state a conjecture about such inequalities, prove it for n less than or equal to 4, and deduce several variants of classical results of Reeve, White and Scarf characterizing the emptiness of small dimensional lattice simplices. For instance, a three dimensional integer simplex is empty if and only if all its faces have width 1. Seemingly different characterizations can be easily proved from one another using the Hermite normal form. In fixed dimension the width of polytopes can be computed in polynomial time (see the simple integer programming formulation of Naase and Ziegler). We prove that it is already NP-complete to decide whether the width of a very special class of integer simplices is 1, and we also provide for every n greater than or equal to 3 a simple example of n-dimensional empty integer simplices of width n - 2, improving on earlier bounds.

关键词： integer programming

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On the Chvatal rank of certain inequalities 1

引用

7th international conference on integer programming and combinatorial optimization (ipco VII)

作者： Hartmann, M Queyranne, M Wang, YG Univ N Carolina Chapel Hill NC 27599 USA Univ British Columbia Vancouver BC V6T 1Z2 Canada PeopleSoft Inc San Mateo CA 94404 USA

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

the Chvatal rank of an inequality ax less than or equal to b with integral components and valid for the integral hull of a polyhedron P, is the minimum number of rounds of Gomory-Chvatal cutting planes needed to obtain the given inequality. the Chvatal rank is st most one if b is the integral part of the optimum value z(a) of the linear program max{ax : x is an element of P}. We show that, contrary to what was stated or implied by other authors, the converse to the latter statement, namely, the Chvatal rank is at least two if b is less than the integral part of z(a), is not true in general. We establish simple conditions for which this implication is valid, and apply these conditions to several classes of facet-inducing inequalities for travelling salesman polytopes.

关键词： Linear programming

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Semidefinite programming methods for the symmetric traveling salesman problem 1

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Cvetković, Dragoš Čangalović, Mirjana Kovačević-Vujčić, Vera University of Belgrade Serbia

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

In this paper the symmetric traveling salesman problem (STSP) is modeled as a problem of discrete semidefinite programming. A class of semidefinite relaxations of STSP model is defined and two variants of a branch-and-bound technique based on this class of relaxations are proposed. the results of preliminary numerical experiments with randomly generated problems are reported. © Springer-Verlag Berlin Heidelberg 1999.

关键词： Branch and bound method

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Market split and basis reduction: Towards a solution of the cornuéjols-dawande instances 1

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Aardal, Karen Bixby, Robert E. Hurkens, Cor A. J. Lenstra, Arjen K. Smeltink, Job W. Department of Computer Science Utrecht University Netherlands Department of Computational and Applied Mathematics Rice University United States Department of Mathematics and Computing Science Eindhoven University of Technology Netherlands Emerging Technology Citibank N.A United States

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

At the ipco VI conference Cornuéjols and Dawande proposed a set of 0-1 linear programming instances that proved to be very hard to solve by traditional methods, and in particular by linear programming based branch-and-bound. they offered these market spliti nstances as a challenge to the integer programming community. the market split problem can be formulated as a system of linear diophantine equations in 0-1 variables. In our study we use the algorithm of Aardal, Hurkens, and Lenstra (1998) based on lattice basis reduction. this algorithm is not restricted to deal with market split instances only but is a general method for solving systems of linear diophantine equations with bounds on the variables. We show computational results from solving both feasibility and optimization versions of the market split instances with up to 7 equations and 60 variables, and discuss various branching strategies and their effect on the number of nodes enumerated. To our knowledge, the largest feasibility and optimization instances solved before have 6 equations and 50 variables, and 4 equations and 30 variables respectively. We also present a probabilistic analysis describing how to compute the probability of generating infeasible market split instances. the formula used by Cornuéjols and Dawande tends to produce relatively many feasible instances for sizes larger than 5 equations and 40 variables. © Springer-Verlag Berlin Heidelberg 1999.

关键词： integer programming

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Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube 1

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Eisenbrand, Friedrich Schulz, Andreas S. Max-Planck-Institut für Informatik Im Stadtwald SaarbrückenD-66123 Germany MIT Sloan School of Management and Operations Research Center E53-361 CambridgeMA02139 United States

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

Gomory’s and Chvátal’s cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. the number of rounds needed to obtain all valid inequalities is known as the Chv´ atal rank of the polyhedron. It is well-known that the Chv´ atal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small;in fact, the rank of any polytope contained in the n-dimensional 0/1-cube is at most 3n2lg n. this improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n3 lgn). Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coefficients is O(n). the latter observation explains why for most cutting planes derived in polyhedral studies of several popular combinatorial optimization problems only linear growth has been observed (see, e.g., [13]);the coefficients of the corresponding inequalities are usually small. Similar results were only known for monotone polyhedra before. Finally, we provide a family of polytopes contained in the 0/1-cube the Chvátal rank of which is at least (1 + Ε)nfor some Ε > 0;the best known lower bound was n. © Springer-Verlag Berlin Heidelberg 1999.

关键词： combinatorial optimization

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