Given a bipartite graph G = (S , T , E ), we consider the problem of finding k bipartite subgraphs, called clusters, such that each vertex i of S appears in exactly one of them, every vertex j of T appears in each cluster in which at least one of its neighbors appears, and the total number of edges needed to make each cluster complete (i.e., to become a biclique) is minimized. This problem is known as k-clustering Minimum Biclique Completion Problem and has been shown strongly NP-hard. It has applications in bundling channels for multicast transmissions. Given a set of demands of services from clients, the application consists of finding k multicast sessions that partition the set of demands. Each service has to belong to a single multicast session, while each client can appear in more sessions. We extend previous work by developing a Branch and Price algorithm that embeds a new metaheuristic based on Variable Neighborhood Infeasible Search and a non-trivial branching rule. The metaheuristic is also adapted to solve efficiently the pricing subproblem. In addition to the random instances used in the literature, we present structured instances generated using the MovieLens data set collected by the GroupLens Research Project. Extensive computational results show that our Branch and Price algorithm outperforms the approaches proposed in the literature.
S. Gualandi, F. Maffioli, C. Magni. A Branch-and-Price Approach to the k-Clustering Minimum Biclique Completion Problem. International Transactions in Operational Research, 2012, Forthcoming.
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