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

Given a graph G=(V,E)G=(V,E), a family of nonempty vertex-subsets S⊆2VS⊆2V, and a weight w:S→R+w:S→R+, the maximum stable set problem with weights on vertex-subsets   consists in finding a stable set II of GG maximizing the sum of the weights of the sets in SS that intersect II. 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.