Tabu Search for the Graph Coloring Problem (extended)

Given an undirected graph G = (V, E), the Graph Coloring Problem (GCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. A subset of V is called a stable set if no two adjacent vertices belong to it. A k coloring of G is a coloring which uses k colors, and corresponds to a partition of V into k stable sets. The Graph Coloring problem is in the original list of NP-hard problems (see Garey and Johnson [8]), and has received a large attention in the literature, not only for its real world applications in many engineering fields, including, among many others, scheduling [10], timetabling [4], register allocation [3], train platforming [2], frequency assignment [7] and communication networks [14], but also for its theoretical aspects and for its difficulty from the computational point of view. Recently, Malaguti et al. [11] proposed an evolutionary algorithm combining a Tabu Search procedure, based on the Impasse Class Neighborhood, and a crossover operator. The Impasse Class Neighborhood was defined by Morgenstern in [13]. It is a structure used to improve a partial k coloring to a complete coloring of the same value, thus, a method which works with a fixed number of colors k and partial colorings (not all vertices are colored). A solution S is a partition of V in k+1 color classes {V1, ..., Vk, Vk+1} in which all classes, but possibly the last one, are stable sets. This means that the first k classes constitute a partial feasible k coloring, while all vertices that do not fit in the first k classes are in the last one. Making this last class empty gives a complete feasible k coloring. To move from a solution S to a new solution S′ in the neighborhood, one can choose an uncolored vertex v ∈ Vk+1, assign v to a different color class, say h, and move to class k + 1 all vertices v′ in class h that are adjacent to v. This ensures that color class h remains feasible. While Morgenstern embedded the neighborhood in a simulated annealing algorithm, Malaguti et al.[11] exploited it in a Tabu search algorithm. The uncolored vertex v is randomly chosen, while the class h is chosen by comparing different target classes by means of an evaluating function f(S). Rather than simply minimizing | Vk+1 |, the algorithm minimizes the global degree of the uncolored vertices: