Models and heuristic algorithms for a weighted vertex coloring problem

We consider a weighted version of the well-known Vertex Coloring Problem (VCP) in which each vertex i of a graph G has associated a positive weight w (i) . Like in VCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used. While in VCP the cost of each color is equal to one, in the Weighted Vertex Coloring Problem (WVCP) the cost of each color depends on the weights of the vertices assigned to that color, and it equals the maximum of these weights. WVCP is known to be NP-hard and arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose three alternative Integer Linear Programming (ILP) formulations for WVCP: one is used to derive, dropping integrality requirement for the variables, a tight lower bound on the solution value, while a second one is used to derive a 2-phase heuristic algorithm, also embedding fast refinement procedures aimed at improving the quality of the solutions found. Computational results on a large set of instances from the literature are reported.