The maximum k-colorable subgraph problem and orbitopes

  • Authors:

  • Tim Januschowski;Marc E. Pfetsch

  • Affiliations:

  • Cork Constraint Computation Centre and Computer Science Department, University College Cork, Cork, Ireland;Institute for Mathematical Optimization, TU Braunschweig, Pockelsstr. 14, 38106 Braunschweig, Germany

  • Venue:
  • Discrete Optimization
  • Year:
  • 2011

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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.