{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8656","name":"Bicovering: Covering Edges With Two Small Subsets of Vertices","abstract":"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.\r\n\r\nGiven 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 [Amb\u00fchl 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.\r\n\r\nOn 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.","keywords":["Bi-covering","Unique Games","Max Bi-clique"],"author":[{"@type":"Person","name":"Bhangale, Amey","givenName":"Amey","familyName":"Bhangale"},{"@type":"Person","name":"Gandhi, Rajiv","givenName":"Rajiv","familyName":"Gandhi"},{"@type":"Person","name":"Hajiaghayi, Mohammad Taghi","givenName":"Mohammad Taghi","familyName":"Hajiaghayi"},{"@type":"Person","name":"Khandekar, Rohit","givenName":"Rohit","familyName":"Khandekar"},{"@type":"Person","name":"Kortsarz, Guy","givenName":"Guy","familyName":"Kortsarz"}],"position":6,"pageStart":"6:1","pageEnd":"6:12","dateCreated":"2016-08-23","datePublished":"2016-08-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bhangale, Amey","givenName":"Amey","familyName":"Bhangale"},{"@type":"Person","name":"Gandhi, Rajiv","givenName":"Rajiv","familyName":"Gandhi"},{"@type":"Person","name":"Hajiaghayi, Mohammad Taghi","givenName":"Mohammad Taghi","familyName":"Hajiaghayi"},{"@type":"Person","name":"Khandekar, Rohit","givenName":"Rohit","familyName":"Khandekar"},{"@type":"Person","name":"Kortsarz, Guy","givenName":"Guy","familyName":"Kortsarz"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6258","volumeNumber":55,"name":"43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)","dateCreated":"2016-08-23","datePublished":"2016-08-23","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"},{"@type":"Person","name":"Rabani, Yuval","givenName":"Yuval","familyName":"Rabani"},{"@type":"Person","name":"Sangiorgi, Davide","givenName":"Davide","familyName":"Sangiorgi"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8656","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6258"}}}