Obtaining the Grundy chromatic number: How bad can my greedy heuristic coloring be?

Abstract

Given a simple undirected graph G, its Grundy chromatic number Γ(G) (or Grundy number) defines the worst-case behavior for the well-known and widely-used greedy first-fit coloring heuristic. Specifically, Γ(G) is the largest k for which a k-coloring can be obtained with the first-fit heuristic. In this paper, we address the Grundy coloring problem, the optimization problem consisting of obtaining the Grundy number of a graph. First, we propose a new combinatorial upper bound for the problem. Second, we describe a standard integer programming formulation and a formulation by representatives. The latter attempts to overcome the symmetries in the problem and relies on the idea that a subset of the vertices in the graph can be represented by one of its vertices, denoted as a representative. Finally, as integer programming approaches can struggle to tackle instances of the problem, we devise a biased random-key genetic algorithm (BRKGA) to obtain heuristic solutions in low computational times. Computational experiments show that our new upper bound can improve over well-established combinatorial bounds available in the literature for several instances. The results also indicate that the formulation by representatives has an overall superior performance than the standard formulation, achieving better results for the denser instances while slightly underperforming on the sparser ones. Furthermore, the BRKGA can find high-quality solutions and can be used with confidence in large instances where the formulations fail.