The maximum-impact coloring polytope

Abstract

Given two graphs G = (V, EG) and H = (V, EH) over the same set of vertices and given a set of colors C, the impact on H of a coloring c : V ! C of G, denoted I(c), is the number of edges ij 2 EH such that c(i) = c(j). In this setting, the maximum-impact coloring problem asks for a proper coloring of G maximizing the impact I(c) on H. This problem naturally arises in the assignment of classrooms to courses, where it is desirable {but not mandatory{ to assign lectures from the same course to the same classroom. Since the maximum-impact coloring problem is NP-hard, we propose in this work an integer-programming-based approach for tackling this problem. To this end, we present an integer programming formulation and we study the associated polytope. We provide several families of valid inequalities, and we study under which conditions these inequalities dene facets of the associated polytope. Finally, we show computational evidence over real-life instances suggesting that some of these families may be useful in a cutting-plane environment.