Hardness of Bipartite Expansion

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 ⊆ ∩ε>0DTIME ( 2nε ) 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]. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems