Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms

The celebrated Cheeger's Inequality [AM85,a86] establishes a bound on the 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 paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. Our operator can be viewed as the gradient operator applied to a certain natural quadratic form for hypergraphs. We show that various hypergraph parameters (for e.g. expansion, diameter, etc) can be bounded using this operator's eigenvalues. We study the heat diffusion process associated with this Laplacian operator, and bound its parameters in terms of its spectra. All our results are generalizations of the corresponding results for graphs. We show that there can be no linear operator for hypergraphs whose spectra captures hypergraph expansion in a Cheeger-like manner. Our Laplacian operator is non-linear, and thus computing its eigenvalues exactly is intractable. For any k, we give a polynomial time algorithm to compute an approximation to the kth smallest eigenvalue of the operator. We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k. Finally, 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.