The maximum [formula omitted]-colorable subgraph problem and orbitopes

Abstract

Given an undirected node-weighted graph and a positive integer k , the maximum k -colorable subgraph problem is to select a k -colorable induced subgraph of largest weight. The natural integer programming formulation for this problem exhibits a high degree of symmetry which arises by permuting the color classes. It is well known that such symmetry has negative effects on the performance of branch-and-cut algorithms. Orbitopes are a polyhedral way to handle such symmetry and were introduced in Kaibel and Pfetsch (2008) [2]. The main goal of this paper is to investigate the polyhedral consequences of combining problem-specific structure with orbitope structure. We first show that the LP-bound of the integer programming formulation mentioned above can only be slightly improved by adding a complete orbitope description. We therefore investigate several classes of facet-defining inequalities for the polytope obtained by taking the convex hull of feasible solutions for the maximum k -colorable subgraph problem that are contained in the orbitope. We study conditions under which facet-defining inequalities for the polytope associated with the maximum k -colorable subgraph problem and the orbitope remain facet-defining for the combined polytope or can be modified to yield facets. It turns out that the results depend on both the structure and the labeling of the underlying graph.