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On Sparsest Cut and Conductance in Directed Polymatroidal Networks

Chandra Chekuri, Anand Louis

TL;DR

A non-constructive Cheeger like inequality that generalizes previous work on hypergraphs is proved and a unified treatment via line-embeddings which were shown to be effective for submodular cuts in ChekuriKRV15 is provided.

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}.

On Sparsest Cut and Conductance in Directed Polymatroidal Networks

TL;DR

A non-constructive Cheeger like inequality that generalizes previous work on hypergraphs is proved and a unified treatment via line-embeddings which were shown to be effective for submodular cuts in ChekuriKRV15 is provided.

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 -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 approximation where 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}.

Paper Structure

This paper contains 45 sections, 38 theorems, 165 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Submodular set functions:
  3. Polymatroidal Networks:
  4. Cut and Partitioning Problems
  5. Conductance:
  6. Submodular and Polymatroidal Hypergraphs:
  7. Our results
  8. Discussion and Related Work
  9. Expansion in (vanilla) hypergraphs.
  10. Vertex expansion in graphs.
  11. Cheeger's inequalities.
  12. Preliminaries and Background
  13. Submodular functions and continuous extensions
  14. Lovász extension:
  15. Base polytope and relation to extensions:
  16. ...and 30 more sections

Key Result

Theorem 1.3

There is a polynomial time $O\left(\sqrt{\log n}\right)$-approximation algorithm for $\Phi_G$ of a directed polymatroidal network $G = \left(V,E,(w_i)_{i \in V},(\rho^+_{v},\rho^-_{v})_{v \in V}\right)$.

Figures (4)

  • Figure 1: LP relaxation for uniform sparsest cut in a directed polymatroidal network
  • Figure 2: SDP Relaxation for uniform sparsest cut in a directed polymatroidal network.
  • Figure 3: SDP Relaxation for symmetric conductance in directed polymatroidal networks.
  • Figure 4: SDP relaxation for the symmetric conductance of polymatroidal hypergraphs.

Theorems & Definitions (78)

  • Definition 1.1: Sparsest cut
  • Definition 1.2: Conductance
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.3
  • Lemma 2.4
  • ...and 68 more