Minimum sum coloring problem: Upper bounds for the chromatic strength

Abstract The minimum sum coloring problem ( M S C P ) is a vertex coloring problem in which a weight is associated with each color. Its aim is to find a coloring of the vertices of a graph G with the minimum sum of the weights of the used colors. The M S C P has important applications in the fields such as scheduling and VLSI design. The minimum number of colors among all optimal solutions of the M S C P for G is called the chromatic strength of G and is denoted by s ( G ) . A tight upper bound of s ( G ) allows to significantly reduce the search space when solving the M S C P . In this paper, we propose and empirically evaluate two new upper bounds of s ( G ) for general graphs, U B A and U B S , based on an abstraction of all possible colorings of G formulated as an ordered set of decreasing positive integer sequences. The experimental results on the standard benchmarks DIMACS and COLOR show that U B A is competitive, and that U B S is significantly tighter than those previously proposed in the literature for 70 graphs out of 74 and, in particular, reaches optimality for 8 graphs. Moreover, both U B A and U B S can be applied to the more general optimum cost chromatic partition ( O C C P ) problem.