Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders

TL;DR

This paper investigates the limitations of the Lasserre hierarchy in approximating dispersers and bipartite expanders, providing strong integrality gaps and an approximation algorithm for these combinatorial structures.

Contribution

It establishes new integrality gaps for the Lasserre hierarchy on dispersers and expanders, and introduces an approximation algorithm matching these gaps.

Findings

Lasserre hierarchy cannot distinguish certain disperser properties in random bipartite graphs.

Existence of infinitely many degrees where the hierarchy fails to distinguish disperser qualities.

An efficient algorithm approximates disperser properties within a ratio matching the integrality gap.

Abstract

We study the problem of approximating the quality of a disperser. A bipartite graph $G$ on $([N],[M])$ is a $(\rho N,(1-\delta)M)$-disperser if for any subset $S\subseteq [N]$ of size $\rho N$, the neighbor set $\Gamma(S)$ contains at least $(1-\delta)M$ distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers. \begin{enumerate} \item For any $\alpha>0$, $\delta>0$, 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 $(N^{\alpha},(1-\delta)M)$-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 random bipartite 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 subset of size exact $\rho N$ that has an approximation ratio matching the integrality gap 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 bipartite expanders with left degree~$D$. $G$ on $([N],[M])$ is a $(\rho N,a)$-expander if 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 that for any constant $\epsilon>0$, there exist constants $\epsilon'<\epsilon,\rho,$ and $D$ such that the Lasserre hierarchy cannot distinguish whether a bipartite graph on $([N],[M])$ with left degree $D$ is a $(\rho N, (1-\epsilon')D)$-expander or not a $(\rho N, (1-\epsilon)D)$-expander.