An Exact Cutting Plane Algorithm to Solve the Selective Graph Coloring Problem in Perfect Graphs

Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph in such a way that no two adjacent vertices share the same color. The Selective Graph Coloring Problem is a generalization of the classical graph coloring problem. Given a graph with a partition of its vertex set into clusters, the aim of the selective graph coloring problem is to pick exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimized. This study focuses on an exact cutting plane algorithm for selective coloring in perfect graphs, where the selective coloring problem is known to be NP-hard. Since there exists no suite of perfect graph instances to the best of our knowledge, we also propose an algorithm to randomly generate perfect graphs and provide a large collection of instances available online. We test our method on graphs with different size and densities, present computational results and compare them with an integer programming formulation of the problem solved by CPLEX, and a state-of-the art algorithm from the literature. Our computational experiments indicate that our approach significantly improves the solution performance.