Spectral hypergraph sparsification via chaining

TL;DR

This paper presents an improved spectral sparsification method for hypergraphs, reducing the number of hyperedges needed while maintaining spectral properties, advancing hypergraph sparsification techniques.

Contribution

It introduces a new spectral hypergraph sparsifier with fewer hyperedges, improving previous bounds and providing a more efficient sparsification approach.

Findings

Achieves spectral $ ext{ε}$-sparsifier with $O( ext{ε}^{-2} ext{log}(D) n ext{log} n)$ hyperedges.

Improves upon previous bounds by Kapralov et al. (2021) and Bansal et al. (2019).

Independent similar results obtained by Jambulapati et al. (2022).

Abstract

In a hypergraph on $n$ vertices where $D$ is the maximum size of a hyperedge, there is a weighted hypergraph spectral $\varepsilon$-sparsifier with at most $O(\varepsilon^{-2} \log(D) \cdot n \log n)$ hyperedges. This improves over the bound of Kapralov, Krauthgamer, Tardos and Yoshida (2021) who achieve $O(\varepsilon^{-4} n (\log n)^3)$, as well as the bound $O(\varepsilon^{-2} D^3 n \log n)$ obtained by Bansal, Svensson, and Trevisan (2019). The same sparsification result was obtained independently by Jambulapati, Liu, and Sidford (2022).