On the complete set packing and set partitioning polytopes: Properties and rank 1 facets

Abstract This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary n -row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets.

[1]  Guy Desaulniers,et al.  New Enhancements for the Exact Solution of the Vehicle Routing Problem with Time Windows , 2016, INFORMS J. Comput..

[2]  Manoel B. Campêlo,et al.  A New Facet Generating Procedure for the Stable Set Polytope , 2011, Electron. Notes Discret. Math..

[3]  Leslie E. Trotter,et al.  On stable set polyhedra for K1, 3-free graphs , 1981, J. Comb. Theory, Ser. B.

[4]  Marcus Poggi de Aragão,et al.  Improved branch-cut-and-price for capacitated vehicle routing , 2016, Mathematical Programming Computation.

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

[6]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[7]  Egon Balas,et al.  Graph substitution and set packing polytopes , 1977, Networks.

[8]  Leslie E. Trotter,et al.  A class of facet producing graphs for vertex packing polyhedra , 1975, Discret. Math..

[9]  Mercedes Landete,et al.  New facets for the set packing polytope , 2000, Oper. Res. Lett..

[10]  Eddie Cheng,et al.  On the Facet-Inducing Antiweb-Wheel Inequalities for Stable Set Polytopes , 2002, SIAM J. Discret. Math..

[11]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[12]  David Pisinger,et al.  Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows , 2008, Oper. Res..

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

[14]  Egon Balas,et al.  Some Valid Inequalities for the Set Partitioning Problem , 1977 .

[15]  Hanif D. Sherali,et al.  Tighter Representations for Set Partitioning Problems , 1996, Discret. Appl. Math..

[16]  Eddie Cheng,et al.  Wheel inequalities for stable set polytopes , 1997, Math. Program..

[17]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra II: Stable Sets , 1994, SIAM J. Discret. Math..

[18]  D. Yun Yeh,et al.  A Dynamic Programming Approach to the Complete Set Partitioning Problem , 1986, BIT.

[19]  L. A. Wolsey,et al.  Further facet generating procedures for vertex packing polytopes , 1976, Math. Program..

[20]  M. Padberg.  On the Complexity of Set Packing Polyhedra , 1977 .

[21]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[22]  Nicholas R. Jennings,et al.  A hybrid exact algorithm for complete set partitioning , 2016, Artif. Intell..

[23]  Marcus Poggi de Aragão,et al.  Limited memory Rank-1 Cuts for Vehicle Routing Problems , 2017, Oper. Res. Lett..

[24]  A. R. Mahjoub,et al.  On the stable set polytope of a series-parallel graph , 1988, Math. Program..

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

[26]  Egon Balas,et al.  Critical Cutsets of Graphs and Canonical Facets of Set Packing Polytopes , 1998 .

[27]  E. Balas,et al.  Set Partitioning: A survey , 1976 .