Spectral Sparsification of Hypergraphs

TL;DR

This paper introduces a polynomial-time algorithm for spectral sparsification of hypergraphs, reducing hyperedges while preserving spectral properties, thereby enabling faster algorithms for related problems and efficient submodular function representation.

Contribution

The paper presents the first polynomial-time method for spectral sparsification of hypergraphs with explicit bounds on hyperedge reduction, improving computational efficiency for spectral and combinatorial tasks.

Findings

Constructs $oldsymbol{ ilde{G}}$ with $O(n^3 ext{log} n/ ext{epsilon}^2)$ hyperedges

Enables faster eigenvalue and cut computations on hypergraphs

Provides a concise hypergraph representation for submodular functions

Abstract

For an undirected/directed hypergraph $G=(V,E)$, its Laplacian $L_G\colon\mathbb{R}^V\to \mathbb{R}^V$ is defined such that its ``quadratic form'' $\boldsymbol{x}^\top L_G(\boldsymbol{x})$ captures the cut information of $G$. In particular, $\boldsymbol{1}_S^\top L_G(\boldsymbol{1}_S)$ coincides with the cut size of $S \subseteq V$, where $\boldsymbol{1}_S \in \mathbb{R}^V$ is the characteristic vector of $S$. A weighted subgraph $H$ of a hypergraph $G$ on a vertex set $V$ is said to be an $\epsilon$-spectral sparsifier of $G$ if $(1-\epsilon)\boldsymbol{x}^\top L_H(\boldsymbol{x}) \leq \boldsymbol{x}^\top L_G(\boldsymbol{x}) \leq (1+\epsilon)\boldsymbol{x}^\top L_H(\boldsymbol{x})$ holds for every $\boldsymbol{x} \in \mathbb{R}^V$. In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph $G$ on $n$ vertices, constructs an $\epsilon$-spectral sparsifier of $G$ with $O(n^3\log n/\epsilon^2)$ hyperedges/hyperarcs. The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of $L_G$, computing the effective resistance between a pair of vertices in $G$, semi-supervised learning based on $L_G$, and cut problems on $G$. In addition, our sparsification result implies that any submodular function $f\colon 2^V \to \mathbb{R}_+$ with $f(\emptyset)=f(V)=0$ can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions $f\colon 2^V \to [0,1]$ with $f(\emptyset)=f(V)=0$, with $O(n^4\log (n/\epsilon) /\epsilon^4)$ samples.