A continuous characterization of the maximum-edge biclique problem

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 graphG, we are able to formulate a continuous optimization problem (namely, an approximate rankone matrix factorization problem with nonnegativity constraints, R1N for short), and show that there is a one-to-one correspondence between (i) the maximum (i.e., the largest) bicliques of G and the global minima of R1N, and (ii) 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.