The Densest k-Subhypergraph Problem

Eden Chlamtác, Michael Dinitz, Christian Konrad, Guy Kortsarz, George Rabanca

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

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Abstract

The Densest k-Subgraph (DkS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V, E), find a subset W subseteq V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4(4-sqrt{3})/13 + epsilon}) <= O(n^{0.697831+epsilon})-approximation (for arbitrary constant epsilon > 0) for Densest k-Subhypergraph and an ~O(n^{2/5})-approximation for Minimum p-Union. We also give an O(sqrt{m})-approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques, APPROX/RANDOM 2016, September 7-9, 2016, Paris, France
EditorsKlaus Jansen, Claire Mathieu, JosE D P Rolim, Chris Umans#
PublisherSchloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany
Pages6:1-6:19
Number of pages19
ISBN (Print)9783959770187
DOIs
Publication statusPublished - 2016

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
Volume60
ISSN (Print)1868-8969

Keywords

  • Hypergraphs
  • Approximation algorithms

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