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Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many p ..."
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Cited by 22 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
EUCLID CALMA Radio Link Frequency Assignment Project Report 2.2.1: Implementation and Testing of Polyhedral Techniques and Interior Point Methods
 Technical Annex T2.2.1 A, T.U. Eindhoven RLFAP Group and T.U. Delft RLFAP Group
, 1995
"... g all the problems of a given class, would need significant theoretical advances. Moreover the choice of an approach is frequently based on several criteria (accuracy, speed, robustness, development cost, etc ) and it is well known that multicriteria based decisions are vey difficult. Nevertheless ..."
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Cited by 2 (0 self)
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g all the problems of a given class, would need significant theoretical advances. Moreover the choice of an approach is frequently based on several criteria (accuracy, speed, robustness, development cost, etc ) and it is well known that multicriteria based decisions are vey difficult. Nevertheless, some empirical rules based mainly on experimental results, but supported in part by theoretical analyses, should provide guidance for the decision makers in their choice of approach for any given problem type. Moreover, one of the most significant outcomes of the CALMA project will be guidance for the efficient development of each of the chosen approaches. 2 Concern 2.1 Global Objectives In this scientific and operational context, the following steps have been proposed as part of the EUCLID program (CEPA6RTP 6.4). 1. To solve a set of problems of the same type by various approaches. 2. To determine, according to the obtained results, the weakness, strengths and po
On the Way to Perfection: Primal Operations for Stable Sets in Graphs
"... In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a pure ..."
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In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a purely combinatorial augmentation procedure for finding a maximum weighted stable set in a graph. Starting with a stable set in a given graph one defines a simplex type tableau whose associated basic feasible solution is the incidence vector of the stable set. In an iterative fashion, nonbasic columns that would lead to pivoting into nonintegral basic feasible solutions, are replaced by new columns that one can read o# from special graph structures such as odd holes, odd antiholes, and various generalizations. Eventually, either a pivot leading to an integral basic feasible solution is performed, or the optimality of the current solution is proved.
A Branch and Cut Algorithm for the Pallet Loading Problem
, 2003
"... We propose a branch and cut algorithm for the Pallet Loading Problem. The 01 formulation proposed by Beasley for cutting problems is adapted to the problem, adding new constraints and new procedures for variable reduction. We then take advantage of the relationship between this problem and the maxi ..."
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We propose a branch and cut algorithm for the Pallet Loading Problem. The 01 formulation proposed by Beasley for cutting problems is adapted to the problem, adding new constraints and new procedures for variable reduction. We then take advantage of the relationship between this problem and the maximum independent set problem to use the partial linear description of its associated polyhedron. Finally, we exploit the specific structure of our problem to define the solution graph and to develop efficient separation procedures. We present computational results for the complete sets Cover I (up to 50 boxes) and Cover II (up to 100 boxes).