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DOI: 10.4230/LIPIcs.ICALP.2016.6
URN: urn:nbn:de:0030-drops-62728
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Bhangale, Amey ; Gandhi, Rajiv ; Hajiaghayi, Mohammad Taghi ; Khandekar, Rohit ; Kortsarz, Guy

Bicovering: Covering Edges With Two Small Subsets of Vertices

LIPIcs-ICALP-2016-6.pdf (0.5 MB)


We study the following basic problem called Bi-Covering. Given a graph G(V, E), find two (not necessarily disjoint) sets A subseteq V and B subseteq V such that A union B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et al., Networks, 2006]. A solution that outputs V,emptyset gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - epsilon ratio algorithm for the problem, for any constant epsilon > 0. Given a bipartite graph, Max-bi-clique is a problem of finding largest k*k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP !subseteq intersection_{epsilon > 0} BPTIME(2^{n^{epsilon}}) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture. On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 + o(1) for Minor Free Graph, 2 - 4*delta/3 for graphs with minimum degree delta*n, 2/(1+delta^2/8) for delta-vertex expander, 8/5 for Split Graphs, 2 - (6/5)*1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.

BibTeX - Entry

  author =	{Amey Bhangale and Rajiv Gandhi and Mohammad Taghi Hajiaghayi and Rohit Khandekar and Guy Kortsarz},
  title =	{{Bicovering: Covering Edges With Two Small Subsets of Vertices}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{6:1--6:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-62728},
  doi =		{10.4230/LIPIcs.ICALP.2016.6},
  annote =	{Keywords: Bi-covering, Unique Games, Max Bi-clique}

Keywords: Bi-covering, Unique Games, Max Bi-clique
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016

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