Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders

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 (N, (1 − δ)M)-disperser or not an (N, 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))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 ρN 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, a)-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 ǫ > 0, there exist constants ǫ < ǫ, ρ, and D such that the Lasserre hierarchy cannot distinguish whether a bipartite graph on ([N ], [M ]) with left degree D is a (ρN, (1 − ǫ)D)-expander or not a (ρN, (1 − ǫ)D)-expander.