Complexity and approximation results for the Min Weighted Coloring problem

A k-coloring of G = (V,E) is a partition S = (S1, . . . , Sk) of the node set V of G into stable sets Si (a stable set is a set of pairwise non adjacent nodes). In the usual case, the objective is to determine a node coloring minimizing k. A natural generalization of this problem is obtained by assigning a strictly positive integer weight w(v) for any node v ∈ V , and defining the weight of stable set S of G as w(S) = max{w(v) : v ∈ S}. Then, the objective is to determine a node coloring S = (S1, . . . , Sk) of G minimizing the quantity w(S) = ∑k i=1 w(Si). This problem is easily shown to be NP-hard ; it suffices to consider w(v) = 1, ∀v ∈ V and MIN WEIGHTED NODE COLORING becomes the classical node coloring problem. Other versions of weighted colorings have been studied in Hassin and Monnot [HAS 05].