A polyhedral study of the maximum stable set problem with weights on vertex-subsets

Abstract

Given a graph G = (V, E), a family of nonempty vertex-subsets S ⊆ 2 V , and a weight w : S → R+, the maximum stable set problem with weights on vertex-subsets consists in finding a stable set I of G maximizing the sum of the weights of the sets in S that intersect I. This problem arises within a natural column generation approach for the vertex coloring problem. In this work we perform an initial polyhedral study of this problem, by introducing a natural integer programming formulation and studying the associated polytope. We address general facts on this polytope including some lifting results, we provide connections with the stable set polytope, and we present three families of facet-inducing inequalities.