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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 $S\subseteq [N]$ 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. \begin{enumerate} \item 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 whether $G$ is an $(N^α,(1-δ)M)$-disperser or not an $(N^{1-α},δM)$-disperser. \item 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 $(ρN, (1-Ω(\frac{1-ρ}{ρd + 1-ρ}))M)$-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 $\frac{\min\{\fracρ{1-ρ},\frac{1-ρ}ρ\}}{\log d}$. \end{enumerate} 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 $S\subseteq [N]$ of size $ρN$, the neighbor set $Γ(S)$ contains at least $a \cdot ρN$ distinct vertices. We prove that for any constant $ε>0$, there exist constants $ε'<ε,ρ,$ and $D$ such that the Lasserre hierarchy cannot distinguish whether a bipartite graph on $([N],[M])$ with left degree $D$ is a $(ρN, (1-ε')D)$-expander or not a $(ρN, (1-ε)D)$-expander.