A branch-and-cut algorithm for partition coloring

Let G = (V,E,Q) be a undirected graph, where V is the set of vertices, E is the set of edges, and Q = {Q1, . . . , Qq} is a partition of V into q subsets. We refer to Q1, . . . , Qq as the components of the partition. The Partition Coloring Problem (PCP) consists of finding a subset V ′ of V with exactly one vertex from each component Q1, . . . , Qq and such that the chromatic number of the graph induced in G by V ′ is minimum. This problem is a generalization of the graph coloring problem. This work presents a branchand-cut algorithm proposed for PCP. An integer programming formulation and valid inequalities are proposed. A tabu search heuristic is used for providing primal bounds. Computational experiments are reported for random graphs and for PCP instances originating from the problem of routing and wavelength assignment in all-optical WDM networks.