A cutting plane algorithm for graph coloring

We present an approach based on integer programming formulations of the graph coloring problem. Our goal is to develop models that remove some symmetrical solutions obtained by color permutations. We study

the problem from a polyhedral point of view and determine some families of facets of the 0/1-polytope associated with one of these integer programming formulations. The theoretical results described here are used to design an efficient Cutting Plane algorithm. © 2007 Elsevier B.V. All rights reserved. Keywords: Graph coloring; Integer programming; Facets of polyhedra; Cutting plane algorithms 1. Introduction The graph coloring problem (GCP) is perhaps one of the most well-known problems in graph theory. This problem arises in many applications such as scheduling, timetabling, electronic bandwidth allocation and sequencing. Despite this practical importance, there are relatively few methods available for solving the problem exactly [2,8,9,13,14]. Given a graph G=(V,E) with n vertices and m edges, a coloring of G is an assignment of colors to each vertex such that the endpoints of any edge have different colors. A (k)-coloring of G is a coloring that uses k colors. The chromatic number of G is the smallest number of colors needed to color G and is denoted by afii9851(G). The coloring problem is to determine afii9851(G) and to find a coloring of G that uses afii9851(G) colors. The coloring problem is known to be NP-hard for arbitrary graphs [7], while...

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