A decomposition approach to solve the selective graph coloring problem in some perfect graph families

Abstract

Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP-hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state-of-the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low-density graphs, and regardless of edge-density in the class of chordal graphs. the number of colors necessary to color the vertices in the selection is minimum. In a graph where each cluster consists of a single vertex, SEL-COL becomes equivalent to the classical graph coloring problem [5]. SEL-COL, or partition coloring as it is alternatively called in the literature, is motivated by the wavelength routing and assignment problem, and was introduced in Li and Simha [17]. There, a telecommunication network, which is a collection of terminal nodes linked by optical fibers capable of carrying a certain number of wavelengths, and a set of source-destination node pairs are given. The problem is concerned with finding the minimum number of distinct wavelengths to assign to each route, that is, paths connecting the given source-destination pairs, such that no two routes that have a common link are assigned the same wavelength. In this setting, the set of all possible routes and pairs of routes possessing a common link correspond to vertices and edges respectively, and groups of routes connecting a given source-destination pair constitute the clusters in the host graph. Then, selection of one vertex per cluster achieves the goal of finding a route between each pair of terminals, and coloring of the selection delivers a proper wavelength to each route.In the standard graph coloring problem, assignments (colors) on the entities (vertices) are done without any regard to alternative choices for them. However, as the example applications reveal, there are cases where entities have their own set of feasible options (clusters) and thus should be allocated one among those alternatives. In such cases, where the basic graph coloring framework fails to suffice, SEL-COL bridges the gap by offering the required flexibility.SEL-COL ha...