Cheeger-type approximation for sparsest $st$-cut

TL;DR

This paper presents a polynomial-time approximation algorithm for the $st$-cut Sparsest-Cut problem with product demands, extending Cheeger's inequality to this setting and achieving an $O( oot{ ext{OPT}})$ approximation.

Contribution

It introduces a new approximation algorithm for the $st$-cut Sparsest-Cut problem with product demands, generalizing Cheeger's inequality and improving previous bounds.

Findings

Achieves $O( oot{ ext{OPT}})$ sparsity approximation for the product-demands case.

Provides an $O( ext{log } n)$-approximation for the general demands setting.

Extends Cheeger's inequality to the $st$-cut Sparsest-Cut problem.

Abstract

We introduce the $st$-cut version the Sparsest-Cut problem, where the goal is to find a cut of minimum sparsity among those separating two distinguished vertices $s,t\in V$. Clearly, this problem is at least as hard as the usual (non-$st$) version. Our main result is a polynomial-time algorithm for the product-demands setting, that produces a cut of sparsity $O(\sqrt{\OPT})$, where $\OPT$ denotes the optimum, and the total edge capacity and the total demand are assumed (by normalization) to be $1$. Our result generalizes the recent work of Trevisan [arXiv, 2013] for the non-$st$ version of the same problem (Sparsest-Cut with product demands), which in turn generalizes the bound achieved by the discrete Cheeger inequality, a cornerstone of Spectral Graph Theory that has numerous applications. Indeed, Cheeger's inequality handles graph conductance, the special case of product demands that are proportional to the vertex (capacitated) degrees. Along the way, we obtain an $O(\log n)$-approximation, where $n=\card{V}$, for the general-demands setting of Sparsest $st$-Cut.