New Approximation Bounds for Small-Set Vertex Expansion

The vertex expansion of the graph is a fundamental graph parameter. Given a graph $G=(V,E)$ and a parameter $\delta \in (0,1/2]$, its $\delta$-Small-Set Vertex Expansion (SSVE) is defined as $$\min_{S : |S| = \delta |V|} \frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} }$$ where $\partial^V(S)$ is the vertex boundary of a set $S$. The SSVE~problem, in addition to being of independent interest as a natural graph partitioning problem, is also of interest due to its connections to the Strong Unique Games problem. We give a randomized algorithm running in time $n^{{\sf poly}(1/\delta)}$, which outputs a set $S$ of size $\Theta(\delta n)$, having vertex expansion at most $$\max\left(O(\sqrt{\phi^* \log d \log (1/\delta)}) , \tilde{O}(d\log^2(1/\delta)) \cdot \phi^* \right),$$ where $d$ is the largest vertex degree of the graph, and $\phi^*$ is the optimal $\delta$-SSVE. The previous best-known guarantees for this were the bi-criteria bounds of $\tilde{O}(1/\delta)\sqrt{\phi^* \log d}$ and $\tilde{O}(1/\delta)\phi^* \sqrt{\log n}$ due to Louis-Makarychev [TOC'16]. Our algorithm uses the basic SDP relaxation of the problem augmented with ${\rm poly}(1/\delta)$ rounds of the Lasserre/SoS hierarchy. Our rounding algorithm is a combination of the rounding algorithms of Raghavendra-Tan [SODA'12] and Austrin-Benabbas-Georgiou [SODA'13]. A key component of our analysis is novel Gaussian rounding lemma for hyperedges which might be of independent interest.