Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm

TL;DR

This paper introduces a simple iterative sampling algorithm that achieves nearly optimal spectral sparsification of directed hypergraphs with size bounds close to theoretical lower limits, extending spectral sparsification techniques.

Contribution

First to provide an $O^*(n^2)$-size spectral sparsifier for weighted directed hypergraphs, matching known lower bounds up to logarithmic factors, and introduces a novel iterative sampling approach.

Findings

Constructed an $O^*(n^2)$-size spectral sparsifier for directed hypergraphs.

Established a lower bound of $oldsymbol{ ext{Omega}(n^2/ ext{epsilon})}$ for general directed hypergraphs.

Presented an iterative sampling algorithm for undirected hypergraphs with improved size bounds.

Abstract

Spectral hypergraph sparsification, an attempt to extend well-known spectral graph sparsification to hypergraphs, has been extensively studied over the past few years. For undirected hypergraphs, Kapralov, Krauthgamer, Tardos, and Yoshida~(2022) have proved an $\varepsilon$-spectral sparsifier of the optimal $O^*(n)$ size, where $n$ is the number of vertices and $O^*$ suppresses the $\varepsilon^{-1}$ and $\log n$ factors. For directed hypergraphs, however, the optimal sparsifier size has not been known. Our main contribution is the first algorithm that constructs an $O^*(n^2)$-size $\varepsilon$-spectral sparsifier for a weighted directed hypergraph. Our result is optimal up to the $\varepsilon^{-1}$ and $\log n$ factors since there is a lower bound of $\Omega(n^2)$ even for directed graphs. We also show the first non-trivial lower bound of $\Omega(n^2/\varepsilon)$ for general directed hypergraphs. The basic idea of our algorithm is borrowed from the spanner-based sparsification for ordinary graphs by Koutis and Xu~(2016). Their iterative sampling approach is indeed useful for designing sparsification algorithms in various circumstances. To demonstrate this, we also present a similar iterative sampling algorithm for undirected hypergraphs that attains one of the best size bounds, enjoys parallel implementation, and can be transformed to be fault-tolerant.