Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders

Abstract

We study the problem of approximating the quality of a disperser. A bipartite graph G on ([N ], [M ]) is a (ρN, (1 − δ)M )-disperser if for any subset S ⊆ [N ] of size ρN , the neighbor set Γ(S) contains at least (1 − δ)M distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers.1. For any α > 0, δ > 0, and a random bipartite graph G with left degree D = O(log N ), we prove that the Lasserre hierarchy cannot distinguish whether2. For any ρ > 0, we prove that there exist infinitely many constants d such that the Lasserre hierarchy cannot distinguish whether a random bipartite graph G with right degree d is a (ρN, (1We also provide an efficient algorithm to find a subset of size exact ρN that has an approximation ratio matching the integrality gap within an extra loss of