A Vector Assignment Approach for the Graph Coloring Problem

Authors:
Takao Ono;Mutsunori Yagiura;Tomio Hirata
Affiliations:
Graduate School of Information Science, Nagoya University, Japan;Graduate School of Information Science, Nagoya University, Japan;Graduate School of Information Science, Nagoya University, Japan
Venue:
Learning and Intelligent Optimization
Year:
2008

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Abstract

We consider the relationship between the graph coloring problem (GCP) and the vector assignment problem (VAP). Given an undirected graph, VAP asks to assign a vector to each vertex so as to maximize the minimum angle between the vectors corresponding to adjacent vertices. We show that any solution to the VAP in the 2-dimensional space, which we call the 2-dimensional VAP (2VAP), gives a feasible coloring, and that such transformation can be computed efficiently. We also show that any optimal solution to 2VAP gives an optimal coloring for GCP. Based on this fact, we propose a heuristic algorithm for GCP, whose search space is the set of solutions for 2VAP. The algorithm is quite simple and can be considered as a variant of the threshold accepting. The experiments show that our algorithm works well for graphs with relatively low degree.