Hardness of bipartite expansion

Subhash Khot, Rishi Saket

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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 β, the goal is to find a subset V′ ⊆ V containing β 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 τ, γ > 0 there is no algorithm which, given a constant β > 0 and a bipartite graph G(U, V, E), runs in polynomial time and decides whether • (YES case) There is a subset S ⊆ V s.t. S| ≥ β |V| satisfying | N(S)| ≤ γ |U|, or • (NO case) Any subset S ⊆ V s.t. |S| ≥ τβ|V| satisfies | N(S)| ≥ (1 - γ|U|, unless NP ⊆ ∩ϵoDTIME (2) 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 [23].

Original languageEnglish (US)
Title of host publication24th Annual European Symposium on Algorithms, ESA 2016
EditorsChristos Zaroliagis, Piotr Sankowski
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770156
StatePublished - Aug 1 2016
Event24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark
Duration: Aug 22 2016Aug 24 2016

Publication series

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


Other24th Annual European Symposium on Algorithms, ESA 2016


  • Bipartite expansion
  • Inapproximability
  • PCP
  • Submodular minimization

ASJC Scopus subject areas

  • Software


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