On Sparsest Cut and Conductance in Directed Polymatroidal Networks

TL;DR

This paper develops algorithms and spectral bounds for sparsest cut and conductance in directed polymatroidal networks, extending previous results on hypergraphs and graphs with new approximation guarantees and inequalities.

Contribution

It introduces a unified approach using line-embeddings to generalize approximation algorithms and Cheeger inequalities for directed polymatroidal networks.

Findings

Achieves an $O(\sqrt{ ext{log} n})$-approximation for sparsest cut.

Provides an $O(\sqrt{OPT ext{log} r})$ approximation for conductance.

Proves a non-constructive Cheeger-like inequality for hypergraphs.

Abstract

We consider algorithms and spectral bounds for sparsest cut and conductance in directed polymatrodal networks. This is motivated by recent work on submodular hypergraphs \cite{Yoshida19,LiM18,ChenOT23,Veldt23} and previous work on multicommodity flows and cuts in polymatrodial networks \cite{ChekuriKRV15}. We obtain three results. First, we obtain an $O(\sqrt{\log n})$-approximation for sparsest cut and point out how this generalizes the result in \cite{ChenOT23}. Second, we consider the symmetric version of conductance and obtain an $O(\sqrt{OPT \log r})$ approximation where $r$ is the maximum degree and we point out how this generalizes previous work on vertex expansion in graphs. Third, we prove a non-constructive Cheeger like inequality that generalizes previous work on hypergraphs. We provide a unified treatment via line-embeddings which were shown to be effective for submodular cuts in \cite{ChekuriKRV15}.