Exact and Heuristic Algorithms for the Partition Coloring Problem

We study the Partition Coloring Problem (PCP), a generalization of the Vertex Coloring Problem where the vertex set is partitioned. The PCP asks to select one vertex for each subset of the partition in such a way that the chromatic number of the induced graph is minimum. We propose a new Integer Linear Programming formulation with an exponential number of variables. To solve this formulation to optimality, we design an effective Branch-and-Price algorithm. We propose and compare several meta-heuristic algorithms capable of finding excellent quality solutions in short computing time. Extensive computational experiments on a benchmark test of instances from the literature show that our Branch-and-Price algorithm, combined with the new meta-heuristic algorithms, is able to outperform the state-of-the-art exact approaches for the PCP.