Discrete relaxations of combinatorial programs

This paper investigates {\em relations\/} among combinatorial optimization problems. To establish such relations we introduce a transformation technique \mbox{---{\em aggregation}---} that allows to relax an integer program by means of another integer program. We prove that various families of prominent inequalities for the acyclic subdigraph problem, the multiple knapsack problem, the max cut, graph, and the clique partitioning problem, the set covering problem, and the set packing problem can be derived and separated in polynomial time in this way. Our technique is algorithmic. It has been implemented and used in a set partitioning code.

[1]  M. R. Rao,et al.  The Steiner tree problem I: Formulations, compositions and extension of facets , 1994, Math. Program..

[2]  Gérard Cornuéjols,et al.  On the 0, 1 facets of the set covering polytope , 1989, Math. Program..

[3]  Laurence A. Wolsey,et al.  Valid inequalities for 0-1 knapsacks and mips with generalised upper bound constraints , 1990, Discret. Appl. Math..

[4]  Manfred W. Padberg.  (1,k)-configurations and facets for packing problems , 1980, Math. Program..

[5]  Egon Balas.  The Asymmetric Assignment Problem and Some New Facets of the Traveling Salesman Polytope on a Directed Graph , 1989, SIAM J. Discret. Math..

[6]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[7]  Matteo Fischetti,et al.  {0, 1/2}-Chvátal-Gomory cuts , 1996, Math. Program..

[8]  William R. Pulleyblank,et al.  Formulations for the stable set polytope of a claw-free graph , 1993, IPCO.

[9]  Laurence A. Wolsey,et al.  Faces for a linear inequality in 0–1 variables , 1975, Math. Program..

[10]  M. R. Rao,et al.  The Steiner tree problem II: Properties and classes of facets , 1994, Math. Program..

[11]  Egon Balas,et al.  The perfectly Matchable Subgraph Polytope of an arbitrary graph , 1989, Comb..

[12]  Ting-Yi Sung,et al.  An analytical comparison of different formulations of the travelling salesman problem , 1991, Math. Program..

[13]  Robert Weismantel,et al.  On the 0/1 knapsack polytope , 1997, Math. Program..

[14]  Laurence A. Wolsey,et al.  Formulations and valid inequalities for the node capacitated graph partitioning problem , 1996, Math. Program..

[15]  Ralf Borndörfer,et al.  Set packing relaxations of some integer programs , 2000, Math. Program..

[16]  Yasuki Sekiguchi,et al.  A note on node packing polytopes on hypergraphs , 1983 .

[17]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[18]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .