Safe lower bounds for graph coloring

Abstract

The best known method for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation 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. © 2011 Springer-Verlag.