Finding Small Sparse Cuts Locally by Random Walk

TL;DR

This paper presents local algorithms based on truncated random walks for finding small sparse cuts in graphs, achieving bicriteria approximations with improved conductance guarantees for sublinear volume sets.

Contribution

It introduces novel local algorithms for small sparse cut detection with bicriteria guarantees, improving conductance bounds using random walk techniques.

Findings

Algorithms run in almost linear time relative to output size.

Achieve conductance bounds of O(√(φ/ε)) and O(√(φ ln(k)/ε)) under certain conditions.

Provide better conductance guarantees for sublinear volume sets.

Abstract

We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G=(V,E) and a parameter k <= |E|, the small sparsest cut problem is to find a subset of vertices S with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem: - If there is a subset U with conductance \phi and vol(U) <= k, then there is a polynomial time algorithm to find a set S with conductance O(\sqrt{\phi/\epsilon}) and vol(S) <= k^{1+\epsilon} for any \epsilon > 1/k. - If there is a subset U with conductance \phi and vol(U) <= k, then there is a polynomial time algorithm to find a set S with conductance O(\sqrt{\phi ln(k)/\epsilon}) and vol(S) <= (1+\epsilon)k for any \epsilon > 2ln(k)/k. These algorithms can be implemented locally using truncated random walk, with running time almost linear to the output size. This provides a local graph partitioning algorithm with a better conductance guarantee when k is sublinear.