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# Hardness of Bipartite Expansion

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## Cite As

Subhash Khot and Rishi Saket. Hardness of Bipartite Expansion. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.55

## 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. In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0 there is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether - (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or - (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless NP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms. We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of Raghavendra and Steurer 2010.
##### Keywords
• inapproximability
• bipartite expansion
• PCP
• submodular minimization

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