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Algorithms 2018, 11(6), 84; https://doi.org/10.3390/a11060084

Efficient Approximation for Restricted Biclique Cover Problems

1
Google Research, New York, NY 10011, USA
2
Department of Computer Science, Brown University, Providence, RI 02912, USA
Work partially done while at Brown University.
*
Author to whom correspondence should be addressed.
Received: 31 March 2018 / Revised: 5 June 2018 / Accepted: 6 June 2018 / Published: 12 June 2018
(This article belongs to the Special Issue Algorithms for Hard Problems: Approximation and Parameterization)
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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 factor or feature that relates the observations in the two sets of nodes connected by the biclique). 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 two 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. View Full-Text
Keywords: biclique cover; approximation algorithms; probabilistic models biclique cover; approximation algorithms; probabilistic models
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Epasto, A.; Upfal, E. Efficient Approximation for Restricted Biclique Cover Problems. Algorithms 2018, 11, 84.

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