Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

Chan, Hubert TH and Louis, Anand and Tang, Zhihao Gavin and Zhang, Chenzi (2018) Spectral Properties of Hypergraph Laplacian and Approximation Algorithms. In: JOURNAL OF THE ACM, 65 (3).

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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 is an element of 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.

Item Type:	Journal Article
Publication:	JOURNAL OF THE ACM
Publisher:	ASSOC COMPUTING MACHINERY, 2 PENN PLAZA, STE 701, NEW YORK, NY 10121-0701 USA
Additional Information:	Copy right of this article belong to ASSOC COMPUTING MACHINERY, 2 PENN PLAZA, STE 701, NEW YORK, NY 10121-0701 USA
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	19 Jun 2018 14:46
Last Modified:	28 Feb 2019 08:49
URI:	http://eprints.iisc.ac.in/id/eprint/60056

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