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Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

Authors Anand Louis , Rameesh Paul , Arka Ray

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Author Details

Anand Louis
  • Indian Institute of Science, Bangalore, India
Rameesh Paul
  • Indian Institute of Science, Bangalore, India
Arka Ray
  • Indian Institute of Science, Bangalore, India

Funding

  • Louis, Anand: Supported in part by SERB Award CRG/2023/002896 and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).
  • Paul, Rameesh: Supported by Prime Minister’s Research Fellowship, India.
  • Ray, Arka: Supported by the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).

Acknowledgements

We would like to thank Madhur Tulsiani, Fernando Jeronimo, and anonymous referees for their helpful comments.

Cite AsGet BibTex

Anand Louis, Rameesh Paul, and Arka Ray. Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.33

Abstract

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to k-uniform hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [Louis, Makarychev - TOC'16; Chan, Louis, Tang, Zhang - JACM'18]. In the second approach, one can view a hypergraph as a simplicial complex and study its various topological properties [Linial, Meshulam - Combinatorica'06; Meshulam, Wallach - RSA'09; Dotterrer, Kaufman, Wagner - SoCG'16; Parzanchevski, Rosenthal - RSA'17] and spectral properties [Kaufman, Mass - ITCS'17; Dinur, Kaufman - FOCS'17; Kaufman, Openheim - STOC'18; Oppenheim - DCG'18; Kaufman, Openheim - Combinatorica'20]. In this work, we attempt to bridge these two directions of study by relating the spectrum of up-down walks and swap walks on the simplicial complex, a downward closed set system, to hypergraph expansion. More precisely, we study the simplicial complex obtained by downward closing the given hypergraph and random walks between its levels X(l), i.e., the sets of cardinality l. In surprising contrast to random walks on graphs, we show that the spectral gap of swap walks and up-down walks between level m and l with 1 < m ⩽ l cannot be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap walks between X(1) and X(k-1) cannot be used to infer any bounds on hypergraph conductance. In contrast, we give a Cheeger-like inequality relating the spectra of walks between level 1 and l for any l ⩽ k to hypergraph expansion. This is a surprising difference between swaps walks and up-down walks! Finally, we also give a construction to show that the well-studied notion of link expansion in simplicial complexes cannot be used to bound hypergraph expansion in a Cheeger-like manner.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Sparse Cuts
  • Random Walks
  • Link Expansion
  • Hypergraph Expansion
  • Simplicial Complexes
  • High Dimensional Expander
  • Threshold Rank

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