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(logN), we prove that the Lasserre hierarchy cannot distinguish whether G is an (Nα, (1 - δ)M)-disperser or not an (N1-α.δM)-disperser. 2. 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, (1 - (1 - ρ)d)M)-disperser or not a (ρN,(1 - ω(1-ρ/ ρd+1-ρ))M)-disperser. We also provide an efficient algorithm to find a subset of size exact pN that has an approximation ratio matching the integrality gap within an extra loss of min{ρ/1-ρ,1-ρ/ ρ}/log d. Our method gives an integrality gap in the Lasserre hierarchy for bipartite expanders with left degree D. G on ([N],[M]) is a (ρN, α)-expander if for any subset S ⊆ [N] of size ρN, the neighbor set Γ(s) contains at least a · ρN distinct vertices. We prove that for any constant ∈ > 0, there exist constants ∈′ > ∈, ρ, and D such that the Lasserre hierarchy cannot distinguish whether a bipartite graph on ([N], [M]) with left degree D is a (ρTV, (1 - ∈′)D)-expa.nder or not a (ρN, (1 - ∈)Z))-expander.