A Column Generation-Based Lower Bound for the Minimum Sum Coloring Problem

The objective of this paper is to derive a tight and efficient lower bound for the minimum sum coloring problem. This NP-hard problem is a variant of the classical graph coloring problem where the objective is to minimize the sum of the colors. A column generation approach is proposed to solve the linear relaxation of a set partition-based formulation. Various enhancements are proposed in order to efficiently obtain attractive columns while avoiding as much as possible the exact solution of the huge number of the NP-hard pricing problems. Experimental results conducted on 42 hard benchmark instances show an average reduction of 89.73% of the gap between the best known lower and upper bounds, including 14 new optimality results.