Currently, the best known tradeoff between approximation ratio and complexity for the {\sc 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，对于稀疏割问题，已知的最佳近似比与复杂性权衡由 [Sherman, FOCS 2009] 中的算法实现：该算法对任意 $\varepsilon\in[\Theta(1/\log n),\Theta(1)]$，利用 $O(n^\varepsilon\log^{O(1)}n)$ 次最大流计算，得到 $O(\sqrt{(\log n)/\varepsilon})$ 近似。其工作原理是采用 [Arora-Kale, JACM 2016] 的乘法权重更新算法求解 [Arora-Rao-Vazirani, STOC 2004] 的半定规划松弛。为实现一个乘法权重更新步骤，Sherman 通过再次应用乘法权重更新来近似求解一个多商品流问题。嵌套的乘法权重更新步骤通过某种“链式”算法解决，该算法结合了多次最大流算法的调用结果。我们提出了一种替代方法，避免求解多商品流问题，转而计算“违反路径”。这简化了 Sherman 的算法，消除了对乘法权重更新嵌套应用的需求，并允许并行化：我们展示了如何利用 $O(n^\varepsilon)$ 个处理器，通过 $O(\log^{O(1)}n)$ 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，并给出一个更简洁的版本及新的分析。