The clique partitioning problem: Facets and patching facets

Abstract

AbstractThe clique partitioning problem (CPP) can be formulated as follows: Given is a complete graph G = (V, E), with edge weights wij ∈ ℝ for all {i, j} ∈ E. A subset A ⊆ E is called a clique partition if there is a partition of V into nonempty, disjoint sets V1,…, Vk, such that each Vp (p = 1,…, k) induces a clique (i.e., a complete subgraph), and A = ∪ {{i, j}|i, j ∈ Vp, i ≠ j}. The weight of such a clique partition A is defined as Σ{i,j}∈A wij. The problem is now to find a clique partition of maximum weight. The clique partitioning polytope P is the convex hull of the incidence vectors of all clique partitions of G. In this paper, we introduce several new classes of facet‐defining inequalities of P. These suffice to characterize all facet‐defining inequalities with right‐hand side 1 or 2. Also, we present a procedure, called patching, which is able to construct new facets by making use of already‐known facet‐defining inequalities. A variant of this procedure is shown to run in polynomial time. Finally, we give limited empirical evidence that the facet‐defining inequalities presented here can be of use in a cutting‐plane approach for the clique partitioning problem. © 2001 John Wiley & Sons, Inc.