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 graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$G$$</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$G$$</EquationSource> </InlineEquation> and the global minima of R1N, and (2) the maximal bicliques of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$G$$</EquationSource> </InlineEquation> (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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$G$$</EquationSource> </InlineEquation>. 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. Copyright Springer Science+Business Media New York 2014