A simpler and parallelizable $O(\sqrt{\log n})$-approximation algorithm for Sparsest Cut

TL;DR

This paper introduces a simplified, parallelizable algorithm for the Sparsest Cut problem that achieves an $O(rac{ ext{sqrt}( ext{log} n)}{ ext{polylog} n})$ approximation ratio with significantly reduced complexity and improved parallelization capabilities.

Contribution

It offers an alternative approach that avoids nested multiplicative weights updates, enabling parallel computation and simplifying Sherman's previous algorithm.

Findings

Achieves $O(rac{ ext{sqrt}( ext{log} n)}{ ext{polylog} n})$ approximation ratio.

Reduces complexity by removing nested MW steps.

Enables parallel computation with $O( ext{log}^{O(1)} n)$ maxflows.

Abstract

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.