The maximum-impact coloring polytope

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∈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 study the associated polytope. We provide several families of valid inequalities, and we study under which conditions these inequalities define 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.