The maximum $k$-colorable subgraph problem and related problems

TL;DR

This paper analyzes the maximum $k$-colorable subgraph problem using semidefinite programming relaxations, exploiting symmetries to improve bounds, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces symmetry-exploiting SDP relaxations for M$k$CS and shows they outperform existing bounds on various test instances.

Findings

Proposed relaxations provide strong bounds for M$k$CS.

Symmetry exploitation simplifies relaxations and improves bounds.

Numerical results outperform existing bounds on most instances.

Abstract

The maximum $k$-colorable subgraph (M$k$CS) problem is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the M$k$CS problem that considers various semidefinite programming relaxations including their theoretical and numerical comparisons. To simplify these relaxations we exploit the symmetry arising from permuting the colors, as well as the symmetry of the given graphs when applicable. We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the M$k$CS problem to derive bounds on the chromatic number of a graph. Our numerical results verify that the proposed relaxations provide strong bounds for the M$k$CS problem, and that those outperform existing bounds for most of the test instances.