Towards a better approximation for sparsest cut?

TL;DR

This paper introduces new algorithms that achieve near-optimal sparsest cut approximations in graphs with specific expansion properties, improving upon previous methods for a broad class of graphs.

Contribution

It presents two novel algorithms for sparsest cut approximation under a new local expansion condition, including a combinatorial approach with Small Set Expander Flows.

Findings

Achieves (1+ε)-approximation for graphs with strong local expansion.

Provides algorithms with runtime 2^{O(r)} poly(n) for the problem.

First to achieve such approximation in this broad class of graphs.

Abstract

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs.