We study polynomial-time approximation algorithms for (edge/vertex) Sparsest Cut and Small Set Expansion in terms of $k$, the number of edges or vertices cut in the optimal solution. Our main results are $\mathcal{O}(\text{polylog}\, k)$-approximation algorithms for various versions in this setting. Our techniques involve an extension of the notion of sample sets (Feige and Mahdian STOC'06), originally developed for small balanced cuts, to sparse cuts in general. We then show how to combine this notion of sample sets with two algorithms, one based on an existing framework of LP rounding and another new algorithm based on the cut-matching game, to get such approximation algorithms. Our cut-matching game algorithm can be viewed as a local version of the cut-matching game by Khandekar, Khot, Orecchia and Vishnoi and certifies an expansion of every vertex set of size $s$ in $\mathcal{O}(\log s)$ rounds. These techniques may be of independent interest. As corollaries of our results, we also obtain an $\mathcal{O}(\log opt)$-approximation for min-max graph partitioning, where $opt$ is the min-max value of the optimal cut, and improve the bound on the size of multicut mimicking networks computable in polynomial time.

翻译：我们研究在最优解切割边或顶点数$k$意义下，（边/顶点）稀疏割与小集合扩张的多项式时间近似算法。我们的主要成果是针对该设定中各种变体问题的$\mathcal{O}(\text{polylog}\, k)$近似算法。我们的技术涉及将样本集概念（Feige 与 Mahdian, STOC'06）从小平衡割推广到一般稀疏割。进而证明如何将这一样本集概念与两种算法相结合——一种基于现有LP舍入框架，另一种基于割匹配游戏的新算法——从而得到上述近似算法。我们的割匹配游戏算法可视为Khandekar、Khot、Orecchia与Vishnoi所提出割匹配游戏的局部版本，并能在$\mathcal{O}(\log s)$轮内验证每个规模为$s$的顶点集的扩张性。这些技术可能具有独立的研究价值。作为本结果的推论，我们还得到了一个针对最小最大图划分问题的$\mathcal{O}(\log opt)$近似算法（其中$opt$为最优割的最小最大值），并改进了多项式时间内可计算的多割模拟网络规模的上界。