The maximum k-colorable subgraph problem and orbitopes

Given an undirected node-weighted graph and a positiveinteger k, the maximum k-colorable subgraph problem is to select ak-colorable induced subgraph of largest weight. The natural integerprogramming formulation for this problem exhibits a high degree ofsymmetry which arises by permuting the color classes. It is well knownthat such symmetry has negative effects on the performance of branch-and-cut algorithms. Orbitopes are a polyhedral way to handle suchsymmetry and were introduced in [19].The main goal of this paper is to investigate the polyhedral con-sequences of combining problem-specific structure with orbitope struc-ture. We first show that the LP-bound of the integer programmingformulation mentioned above can only be slightly improved by adding acomplete orbitope description. We therefore investigate several classesof facet-defining inequalities for the polytope obtained by taking theconvex hull of feasible solutions for the maximum k-colorable subgraphproblem that are contained in the orbitope. We study conditions underwhich facet-defining inequalities for the polytope associated with themaximum k-colorable subgraph problem and the orbitope remain facet-defining for the combined polytope or can be modified to yield facets. Itturns out that the results depend on both the structure and the labelingof the underlying graph.