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Algorithms 2018, 11(1), 10; https://doi.org/10.3390/a11010010

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94709, USA
An extended abstract of this work appeared at ICALP 2017 under a different title “Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis”.
Received: 28 November 2017 / Revised: 29 December 2017 / Accepted: 5 January 2018 / Published: 17 January 2018
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Abstract

The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions. View Full-Text
Keywords: hardness of approximation; small set expansion hypothesis; maximum edge biclique; maximum balanced biclique; minimum k-cut; densest at-least-k-subgraph hardness of approximation; small set expansion hypothesis; maximum edge biclique; maximum balanced biclique; minimum k-cut; densest at-least-k-subgraph
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Manurangsi, P. Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis. Algorithms 2018, 11, 10.

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