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We study the problem of approximating the quality of a disperser. A bipartitegraph $G$ on $([N],[M])$ is a $(\rho N,(1-\delta)M)$-disperser if for anysubset $S\subseteq [N]$ of size $\rho N$, the neighbor set $\Gamma(S)$ containsat least $(1-\delta)M$ distinct vertices. Our main results are strongintegrality gaps in the Lasserre hierarchy and an approximation algorithm fordispersers.  \begin{enumerate}  \item For any $\alpha>0$, $\delta>0$, and a random bipartite graph $G$ withleft degree $D=O(\log N)$, we prove that the Lasserre hierarchy cannotdistinguish whether $G$ is an $(N^{\alpha},(1-\delta)M)$-disperser or not an$(N^{1-\alpha},\delta M)$-disperser.  \item For any $\rho>0$, we prove that there exist infinitely many constants$d$ such that the Lasserre hierarchy cannot distinguish whether a randombipartite graph $G$ with right degree $d$ is a $(\rho N,(1-(1-\rho)^d)M)$-disperser or not a $(\rho N, (1-\Omega(\frac{1-\rho}{\rho d +1-\rho}))M)$-disperser. We also provide an efficient algorithm to find a subsetof size exact $\rho N$ that has an approximation ratio matching the integralitygap within an extra loss of$\frac{\min\{\frac{\rho}{1-\rho},\frac{1-\rho}{\rho}\}}{\log d}$.\end{enumerate}  Our method gives an integrality gap in the Lasserre hierarchy for bipartiteexpanders with left degree~$D$. $G$ on $([N],[M])$ is a $(\rho N,a)$-expanderif for any subset $S\subseteq [N]$ of size $\rho N$, the neighbor set$\Gamma(S)$ contains at least $a \cdot \rho N$ distinct vertices. We prove thatfor any constant $\epsilon>0$, there exist constants $\epsilon'<\epsilon,\rho,$and $D$ such that the Lasserre hierarchy cannot distinguish whether a bipartitegraph on $([N],[M])$ with left degree $D$ is a $(\rho N,(1-\epsilon')D)$-expander or not a $(\rho N, (1-\epsilon)D)$-expander.

Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders