Facet-generating Procedures for 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 ∈ EH such that c(i) = c(j). In this setting, the maximum-impact coloring problem asks for a proper coloring c of G maximizing the impact I(c) on H. This problem naturally arises in the context of assigning classrooms to courses, where it is desirable –but not mandatory– to assign lectures from the same course to the same classroom. We are interested in an integer programming approach for this problem. In this work we present two procedures that construct valid inequalities from existing inequalities, based on extending individual colors to sets of colors and on extending edges of G to cliques, respectively. If the original inequality defines a facet and additional technical hypotheses are satisfied, then the obtained inequality also defines a facet. We present a generic separation algorithm based on these procedures, and report computational experiments showing that this approach is effective.