{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8885","name":"Hardness of Bipartite Expansion","abstract":"We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U,V,E) and a parameter beta, the goal is to find a subset V' subseteq V containing beta fraction of the vertices of V which minimizes the size of N(V'), the neighborhood of V'. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion. \r\n\r\nIn this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0\r\nthere is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether \r\n \r\n- (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or \r\n \r\n- (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless \r\nNP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms.\r\n\r\nWe note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of\r\nRaghavendra and Steurer 2010.","keywords":["inapproximability","bipartite expansion","PCP","submodular minimization"],"author":[{"@type":"Person","name":"Khot, Subhash","givenName":"Subhash","familyName":"Khot"},{"@type":"Person","name":"Saket, Rishi","givenName":"Rishi","familyName":"Saket"}],"position":55,"pageStart":"55:1","pageEnd":"55:17","dateCreated":"2016-08-18","datePublished":"2016-08-18","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Khot, Subhash","givenName":"Subhash","familyName":"Khot"},{"@type":"Person","name":"Saket, Rishi","givenName":"Rishi","familyName":"Saket"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ESA.2016.55","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6260","volumeNumber":57,"name":"24th Annual European Symposium on Algorithms (ESA 2016)","dateCreated":"2016-08-18","datePublished":"2016-08-18","editor":[{"@type":"Person","name":"Sankowski, Piotr","givenName":"Piotr","familyName":"Sankowski"},{"@type":"Person","name":"Zaroliagis, Christos","givenName":"Christos","familyName":"Zaroliagis"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8885","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":"#volume6260"}}}