The sum coloring problem: a memetic algorithm based on two individuals

Let G be a graph, for which each vertex is assigned one of k colors in {1,..., k}. A legal coloring requires that two adjacent vertices have two different colors. The minimum sum coloring problem (MSCP) consists of finding such a legal coloring with the smallest sum value, knowing that to each color is assigned an integer value from 1 to k. This paper presents a new memetic approach for this NP-hard problem. The proposed hybrid algorithm is based on a very small population, composed of only two individuals. Thanks to this small population, no selection operator needs to be defined, nor any replacement strategy. In order to prevent a premature convergence of the algorithm, alternative mechanisms are introduced based on an elitist approach. The local search introduced in the population based algorithm is split in two phases. The first one is based on the well known TabuCol algorithm and aims to reduce the number of conflicting vertices. The second one, which is used to improve the total sum value, implements an efficient 2-move local operator. The proposed approach was successfully applied on many graphs from the reference COLORS02 and DIMACS benchmarks. It allowed to achieve most of the best known results, and to overpass the best known results for 15 challenging graphs.