A Branch-and-price Approach to the k-Clustering Minimum Biclique Completion Problem

Given a bipartite graph G = (S, T, E), the k-clustering Minimum Biclique Completion Problem (k-MinBCP) consists of finding k bipartite subgraphs (clusters), such that each vertex i of S appears in exactly one subgraph, every vertex j in T appears in each cluster in which at least one of its neighbors appears, and the total number of edges that would complete each subgraph into a complete bipartite subgraph, i.e., a biclique, is minimized. This problem was introduced in [1], as an application of the problem of bundling channels in multicast transmissions. k-MinBCP is NP-Hard, and its approximability, to the best of our knowledge, remains unknown. In the literature, k-MinBCP is tackled with two approaches: in [1], it is solved with an Integer Programming approach, a Bilinear Programming formulation and its standard linearization; and in [2], it is solved with an hybrid Constraint Programming–Semidefinite programming approach.