Separating stable sets in claw-free graphs via Padberg-Rao and compact linear programs

In this paper, we provide the first linear programming formulations for the stable set problem in claw-free graphs, together with polynomial time separation routines for those formulations (they are not compact). We then exploit one of those extended formulations and propose a new polytime algorithm for solving the separation problem for the stable set polytope of claw-free graphs. This routine combines a separation algorithm for the matching polytope due to Padberg and Rao and the solution of (moderate size) compact linear programs. Hence, it does not rely on the ellipsoid method and seems to be appropriate to be inserted in branch and cut frameworks for solving real world problems.

[1]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[2]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[3]  A. Tamura,et al.  A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph , 2001 .

[4]  Xueliang Li,et al.  A Combinatorial Algorithm for Minimum Weighted Colorings of Claw-Free Perfect Graphs , 2005, J. Comb. Optim..

[5]  Paul D. Seymour,et al.  The structure of claw-free graphs , 2005, BCC.

[6]  Egon Balas,et al.  Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization , 2005, Ann. Oper. Res..

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

[8]  Martin Grötschel,et al.  A cutting plane algorithm for minimum perfect 2-matchings , 1987, Computing.

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

[10]  Gianpaolo Oriolo,et al.  An algorithmic decomposition of claw-free graphs leading to an O(n3)-algorithm for the weighted stable set problem , 2011, SODA '11.

[11]  Egon Balas,et al.  programming: Properties of the convex hull of feasible points * , 1998 .

[12]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[13]  George J. Minty,et al.  On maximal independent sets of vertices in claw-free graphs , 1980, J. Comb. Theory B.

[14]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

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

[16]  Gianpaolo Oriolo,et al.  A New Algorithm for the Maximum Weighted Stable Set Problem in Claw-Free Graphs , 2008, IPCO.

[17]  Volker Kaibel,et al.  Branched Polyhedral Systems , 2010, IPCO.