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Parallel Combinatorial Optimization Team Parallel Branch-and-Bound Algorithms using Semidef.pdf

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Parallel Combinatorial Optimization Team Parallel Branch-and-Bound Algorithms using Semidef

Parallel Branch-and-Bound Algorithms usingSemide nite Programming Relaxation for theIndependent Set ProblemPractical Training ReportMay-July 1999 date: October 16, 2000. byW.J. van Hoeve University of TwenteEnschede, the NetherlandsFaculty of Mathematical SciencesDepartment of Discrete Mathematics, OperationsResearch and StatisticsOptimization Modeling GroupPRiSM Laboratory - CNRS UMR 8636University of Versailles - Saint Quentin en YvelinesVersailles, FranceParallel Combinatorial Optimization Team Parallel Branch-and-Bound Algorithms usingSemide nite Programming Relaxation for theIndependent Set ProblemPractical Training ReportMay-July 1999 date: October 16, 2000. byW.J. van Hoeve University of TwenteEnschede, the NetherlandsFaculty of Mathematical SciencesDepartment of Discrete Mathematics, OperationsResearch and StatisticsOptimization Modeling GroupPRiSM Laboratory - CNRS UMR 8636University of Versailles - Saint Quentin en YvelinesVersailles, FranceParallel Combinatorial Optimization TeamSupervisors:C. Roucairol (PRiSM)V.-D. Cung (PRiSM)F. Roupin (CEDRIC-IIE)S.M. Baas (UT) SummaryThis report contains a study on the weighted independent set problem. In partic-ular, it deals with solving this problem by parallel branch-and-bound algorithms,in which the evaluation of the nodes is based on positive semide nite program-ming relaxation.There have been evaluated two dierent positive semide nite programs. Bothprograms provide an upper bound on the independent set problem, and are wellsuited for the use inside a branch-and-bound algorithm.Comparisons with a standard linear LP relaxation show that the bounds com-puted with the positive semide nite programs are much tighter than the boundcomputed with the standard LP relaxation program.The semide nite program that provides the tightest bounds, also needs thehighest computation time. This computation time increases as the edge densityincreases. The LP relaxation needs less computation time, but here also holds, asthe edge de