New Integer Linear Programming Models for the Vertex Coloring Problem

• Published in: LATIN 2018: Theoretical Informatics Vol. 10807; pp. 640 - 652
• Main Authors: Jabrayilov, Adalat, Mutzel, Petra
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2018; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Graph coloring; Integer linear programming; Vertex coloring
• ISBN: 9783319774039; 3319774034
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-77404-6_47

The vertex coloring problem asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two neighbors have different colors. The problem is NP-hard. Here, we introduce new integer linear programming formulations based on partial-ordering. They have the advantage that they are as simple to work with as the classical assignment formulation, since they can be fed directly into a standard integer linear programming solver. We evaluate our new models using Gurobi and show that our new simple approach is a good alternative to the best state-of-the-art approaches for the vertex coloring problem. In our computational experiments, we compare our formulations with the classical assignment formulation and the representatives formulation on a large set of benchmark graphs as well as randomly generated graphs of varying size and density. The evaluation shows that the partial-ordering based models dominate both formulations for sparse graphs, while the representatives formulation is the best for dense graphs.