Safe lower bounds for graph coloring

Authors:
Stephan Held;William Cook;Edward C. Sewell
Affiliations:
University of Bonn, Georgia Institute of Technology, Southern Illinois University Edwardsville;University of Bonn, Georgia Institute of Technology, Southern Illinois University Edwardsville;University of Bonn, Georgia Institute of Technology, Southern Illinois University Edwardsville
Venue:
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Year:
2011

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Abstract

The best known method for determining lower bounds on the vertex coloring number of a graph is the linear-programming columngeneration technique first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically safe results, independent of the floating-point accuracy of linear-programming software. Our work includes an improved branch-and-bound algorithm for maximum-weight stable sets and a parallel branch-and-price framework for graph coloring. Computational results are presented on a collection of standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.