research-article

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

Authors:

T.-H. Hubert Chan,

Zhihao Gavin Tang,

Chenzi ZhangAuthors Info & Claims

Journal of the ACM (JACM), Volume 65, Issue 3

Article No.: 15, Pages 1 - 48

https://doi.org/10.1145/3178123

Published: 05 March 2018 Publication History

Abstract

The celebrated Cheeger’s Inequality (Alon and Milman 1985; Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph.

In this article, we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger’s Inequality that relates hyperedge expansion with the “second eigenvalue” of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework.

Since higher-order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher-order Cheeger-like inequalities. For any k ∈ N, we give a polynomial-time algorithm to compute an O(log r)-approximation to the kth procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by Raghavendra and Steurer (2010)) for constant values of k.

Moreover, using the factor-preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

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Index Terms

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms

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cover image Journal of the ACM

Journal of the ACM Volume 65, Issue 3

June 2018

285 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3191817

Editor:
Éva Tardos
Cornell University

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Copyright © 2018 ACM.

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Publication History

Published: 05 March 2018

Accepted: 01 January 2018

Revised: 01 November 2017

Received: 01 May 2016

Published in JACM Volume 65, Issue 3

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Hong Kong RGC
Santosh Vempala's NSF
Princeton University
Simons Collaboration on Algorithms and Geometry

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https://doi.org/10.21099/tkbjm/20244801131
Kamal RBagchi A(2024)A Lovász-Simonovits Theorem for Hypergraphs with Application to Local ClusteringProceedings of the ACM on Management of Data10.1145/36771262:4(1-27)Online publication date: 30-Sep-2024
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