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Journal of Global Optimization

, Volume 58, Issue 3, pp 439–464 | Cite as

A continuous characterization of the maximum-edge biclique problem

  • Nicolas Gillis
  • François Glineur
Article

Abstract

The problem of finding large complete subgraphs in bipartite graphs (that is, bicliques) is a well-known combinatorial optimization problem referred to as the maximum-edge biclique problem (MBP), and has many applications, e.g., in web community discovery, biological data analysis and text mining. In this paper, we present a new continuous characterization for MBP. Given a bipartite graph \(G\), we are able to formulate a continuous optimization problem (namely, an approximate rank-one matrix factorization problem with nonnegativity constraints, R1N for short), and show that there is a one-to-one correspondence between (1) the maximum (i.e., the largest) bicliques of \(G\) and the global minima of R1N, and (2) the maximal bicliques of \(G\) (i.e., bicliques not contained in any larger biclique) and the local minima of R1N. We also show that any stationary points of R1N must be close to a biclique of \(G\). This allows us to design a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1N. We show that this algorithm, whose algorithmic complexity per iteration is proportional to the number of edges in the graph, is guaranteed to converge to a biclique and that it performs competitively with existing methods on random graphs and text mining datasets. Finally, we show how R1N is closely related to the Motzkin–Strauss formalism for cliques.

Keywords

Maximum-edge biclique problem Biclique finding algorithm  Algorithmic complexity Nonnegative rank-one approximation 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.ICTEAM InstituteUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Center for Operations Research and EconometricsUniversité catholique de LouvainLouvain-la-NeuveBelgium

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