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.

翻译：图的顶点扩展是一个基本的图参数。给定图 $G=(V,E)$ 和参数 $\delta \in (0,1/2]$，其 $\delta$-小集顶点扩展（SSVE）定义为 \[ \min_{S : |S| = \delta |V|} \frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} } \] 其中 $\partial^V(S)$ 是集合 $S$ 的顶点边界。SSVE问题除了作为自然图划分问题具有独立研究意义外，还因其与强唯一博弈问题的联系而备受关注。我们提出一种随机算法，运行时间为 $n^{{\sf poly}(1/\delta)}$，输出大小为 $\Theta(\delta n)$ 的集合 $S$，其顶点扩展至多为 \[ \max\left(O(\sqrt{\phi^* \log d \log (1/\delta)}) , \tilde{O}(d\log^2(1/\delta)) \cdot \phi^* \right), \] 其中 $d$ 是图的最大顶点度数，$\phi^*$ 是最优 $\delta$-SSVE值。此前该问题的最佳已知保证是 Louis-Makarychev [TOC'16] 给出的 $\tilde{O}(1/\delta)\sqrt{\phi^* \log d}$ 和 $\tilde{O}(1/\delta)\phi^* \sqrt{\log n}$ 双准则界。我们的算法使用该问题的基本SDP松弛，并增加 ${\rm poly}(1/\delta)$ 轮Lasserre/SoS层级。我们的舍入算法结合了 Raghavendra-Tan [SODA'12] 和 Austrin-Benabbas-Georgiou [SODA'13] 的舍入算法。分析的关键组成部分是一个新颖的超边高斯舍入引理，该引理可能具有独立研究价值。