Efficient Approximation for Restricted Biclique Cover Problems

Algorithms, vol. 11(6) (2018)

Abstract

Covering the edges of a bipartite graph by a minimum set of bipartite complete graphs (bicliques) is a basic graph theoretic problem, with numerous applications. In particular, it is used to characterize parsimonious models of a set of observations (each biclique corresponds to a {\it factor} or {\it feature} that relates the observations in the two sets of nodes connected by the biclique). \revision{The decision version of the minimum biclique cover problem is NP-Complete}, and unless $P=NP$, the cover size cannot be approximated in general within less than a sub-linear factor of the number of nodes (or edges) in the graph. In this work, we consider two natural restrictions to the problem, motivated by practical applications. In the first case, we restrict the number of bicliques a node can belong to. We show that when this number is at least $5$, the problem is still NP-hard. In contrast, we show that when nodes belong to no more than 2 bicliques, the problem has efficient approximations. The second model we consider corresponds to observing a set of independent samples from an unknown model, governed by a possibly large number of factors. The model is defined by a bipartite graph $G=(L,R,E)$, where each node in $L$ is assigned to an arbitrary subset of up to a constant $f$ factors, while the nodes in $R$ (the independent observations) are assigned to random subsets of the set of $k$ factors where $k$ can grow with size of the graph. We show that this practical version of the biclique cover problem is amenable to efficient approximations.