Improved Lower Bounds for Sum Coloring via Clique Decomposition

Given an undirected graph $G = (V,E)$ with a set $V$ of vertices and a set $E$ of edges, the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of $G$, using colors represented by natural numbers $1, 2, . . .$ such that the total sum of the colors assigned to the vertices is minimized. This paper describes an approach based on the decomposition of the original graph into disjoint cliques for computing lower bounds for the MSCP. Basically, the proposed approach identifies and removes at each extraction iteration a maximum number of cliques of the same size (the largest possible) from the graph. Computational experiments show that this approach is able to improve on the current best lower bounds for 14 benchmark instances, and to prove optimality for the first time for 4 instances. We also report lower bounds for 24 more instances for which no such bounds are available in the literature. These new lower bounds are useful to estimate the quality of the upper bounds obtained with various heuristic approaches.