Currently, the best known tradeoff between approximation ratio and complexity for the Sparsest Cut problem is achieved by the algorithm in [Sherman, FOCS 2009]: it computes $O(\sqrt{(\log n)/\varepsilon})$-approximation using $O(n^\varepsilon\log^{O(1)}n)$ maxflows for any $\varepsilon\in[\Theta(1/\log n),\Theta(1)]$. It works by solving the SDP relaxation of [Arora-Rao-Vazirani, STOC 2004] using the Multiplicative Weights Update algorithm (MW) of [Arora-Kale, JACM 2016]. To implement one MW step, Sherman approximately solves a multicommodity flow problem using another application of MW. Nested MW steps are solved via a certain ``chaining'' algorithm that combines results of multiple calls to the maxflow algorithm. We present an alternative approach that avoids solving the multicommodity flow problem and instead computes ``violating paths''. This simplifies Sherman's algorithm by removing a need for a nested application of MW, and also allows parallelization: we show how to compute $O(\sqrt{(\log n)/\varepsilon})$-approximation via $O(\log^{O(1)}n)$ maxflows using $O(n^\varepsilon)$ processors. We also revisit Sherman's chaining algorithm, and present a simpler version together with a new analysis.

翻译：目前，稀疏割问题的近似比与复杂度之间的最佳已知权衡由[Sherman, FOCS 2009]中的算法实现：对于任意$\varepsilon\in[\Theta(1/\log n),\Theta(1)]$，该算法使用$O(n^\varepsilon\log^{O(1)}n)$次最大流计算，得出$O(\sqrt{(\log n)/\varepsilon})$近似。其核心是通过[Arora-Kale, JACM 2016]的乘性权重更新算法求解[Arora-Rao-Vazirani, STOC 2004]的半定规划松弛。在实现单次乘性权重更新步骤时，Sherman通过另一乘性权重更新算法近似求解多商品流问题。嵌套的乘性权重更新步骤通过一种结合多次最大流算法结果的“链式”算法完成。我们提出一种替代方法，避免求解多商品流问题，转而计算“违反路径”。这简化了Sherman算法，消除了嵌套乘性权重更新的需求，同时支持并行化：我们展示了如何通过$O(n^\varepsilon)$个处理器使用$O(\log^{O(1)}n)$次最大流计算，得到$O(\sqrt{(\log n)/\varepsilon})$近似。此外，我们重新审视了Sherman的链式算法，并给出简化版本及新分析。