Redundant constraints in the standard formulation for the clique partitioning problem

Abstract

The clique partitioning problem is among the most fundamental graph partitioning problems. Given a complete weighted undirected graph, we find a vertex-disjoint union of cliques so as to maximize the sum of the weights within the cliques. The problem is known to be NP-hard, and exact algorithms and heuristics have been developed so far. In 1989, Grotschel and Wakabayashi (Math Program 45:59–96, 1989) proposed an integer linear programming formulation for the problem. Although this formulation has been employed by many algorithms, it is prohibitive even for relatively small graphs because it contains numerous constraints called transitivity constraints. In this study, we theoretically derive a certain class of redundant constraints in the formulation. Furthermore, we show that the constraints in the derived class are also redundant in the linear programming relaxation of the formulation. By using our results, the number of constraints treated in exact algorithms and heuristics may be reduced. We confirmed that more than half of the transitivity constraints are redundant for some instances, and that computation times to find an optimal solution are shortened for most instances.