A Cheeger Cut for Uniform Hypergraphs

TL;DR

This paper extends the Cheeger constant and inequalities from graphs to uniform hypergraphs, establishing spectral bounds and methods for approximating optimal cuts in hypergraph structures.

Contribution

It introduces a generalized Cheeger constant for uniform hypergraphs and relates it to the eigenvalues of a normalized Laplacian, providing new spectral tools.

Findings

Eigenvalues bound the Cheeger constant in hypergraphs

Eigenfunctions approximate Cheeger cuts in hypergraphs

Spectral methods extend graph partitioning to hypergraphs

Abstract

The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is bounded both above and below by the generalized Cheeger constant, and the corresponding eigenfunctions can be used to approximate the Cheeger cut.