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On Sparsest Cut and Conductance in Directed Polymatroidal Networks (2410.20525v1)

Published 27 Oct 2024 in cs.DS

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}.

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