Near-optimal Linear Sketches and Fully-Dynamic Algorithms for Hypergraph Spectral Sparsification

TL;DR

This paper introduces a simplified framework for hypergraph spectral sparsification that reduces the problem to effective resistance sampling in ordinary graphs, leading to nearly-linear time algorithms and improvements in multiple computational models.

Contribution

The authors present a new framework that simplifies hypergraph spectral sparsification, enabling nearly-linear time algorithms and advancing the state-of-the-art in linear sketching, dynamic, and online models.

Findings

Developed a simpler hypergraph sparsification framework bypassing weight assignment computation.

Achieved a nearly-linear time algorithm for spectral hypergraph sparsification.

Provided the first nearly-optimal algorithms in linear sketching, fully dynamic, and online models.

Abstract

A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size hypergraph spectral sparsifiers. However, the task of computing such sparsifiers turns out to be much more involved, and all known algorithms rely on the notion of balanced weight assignments, whose computation inherently relies on repeated, complete access to the underlying hypergraph. We introduce a significantly simpler framework for hypergraph spectral sparsification which bypasses the need to compute such weight assignments, essentially reducing hypergraph sparsification to repeated effective resistance sampling in \textit{ordinary graphs}, which are obtained by \textit{oblivious vertex-sampling} of the original hypergraph. Our framework immediately yields a simple, new nearly-linear time algorithm for nearly-linear size spectral hypergraph sparsification. Furthermore, as a direct consequence of our framework, we obtain the first nearly-optimal algorithms in several other models of computation, namely the linear sketching, fully dynamic, and online settings.