The biclique partitioning polytope

Abstract

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.