Solving Vertex Coloring Problems as Maximum Weight Stable Set Problems

Denis Cornaz(denis.cornazdauphine.fr)
Fabio Furini(fabio.furinidauphine.fr)
Enrico Malaguti(enrico.malagutiunibo.it)

Abstract: In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by Cornaz and Jost [Oper Res Lett, 36, 2008], where given a graph G, an auxiliary graph G' is constructed, such that the family of all stable sets of G' is in one-to-one correspondence with the family of all feasible colorings of G. The transformation in was originally proposed for the classical Vertex Coloring and the Max-Coloring problems; we extend it to the Equitable Coloring Problem and the Bin Packing Problem with Conflicts. We report extensive computational experiments on benchmark instances of the four problems, and compare the solution method with the state-of-the-art algorithms. By exploiting the proposed method, we largely outperform the state-of-the-art algorithm for the Max-coloring Problem, and we are able to solve, for the first time to proven optimality, 14 Max-coloring and 2 Equitable Coloring instances.

Keywords: Vertex Coloring, Stable Set, Mixed Integer Linear Programming, Computational Experiments

Category 1: Integer Programming

Category 2: Combinatorial Optimization (Graphs and Matroids )

Citation: Tech.Rep. OR-15-5 DEI University of Bologna

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Entry Submitted: 06/19/2015
Entry Accepted: 06/19/2015
Entry Last Modified: 06/19/2015

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