Breaking the Multicommodity Flow Barrier for sqrt(log(n))-Approximations to Sparsest Cut

TL;DR

This paper introduces a new algorithm that achieves a near-optimal approximation for the sparsest cut problem while running in sub-quadratic time by leveraging a novel multicommodity flow approach and a strengthened structure theorem.

Contribution

It presents a combined approach that improves approximation ratios for sparsest cut with faster algorithms using a specialized multicommodity flow technique.

Findings

Achieves O(sqrt(log(n)/eps))-approximation in O(n^eps log^O(1) n) max-flows.

Develops a stronger, algorithmic version of Arora et al.'s structure theorem.

Shows limitations of the cut-matching game framework without flow rerouting.

Abstract

This paper ties the line of work on algorithms that find an O(sqrt(log(n)))-approximation to the sparsest cut together with the line of work on algorithms that run in sub-quadratic time by using only single-commodity flows. We present an algorithm that simultaneously achieves both goals, finding an O(sqrt(log(n)/eps))-approximation using O(n^eps log^O(1) n) max-flows. The core of the algorithm is a stronger, algorithmic version of Arora et al.'s structure theorem, where we show that matching-chaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks. By using that specialized algorithm in place of a black-box solver, we are able to solve those instances much more efficiently. We also show the cut-matching game framework can not achieve an approximation any better than Omega(log(n)/log log(n)) without re-routing flow.