Fully Dynamic Spectral Sparsification for Directed Hypergraphs

TL;DR

This paper introduces a fully dynamic algorithm for maintaining spectral sparsifiers of directed hypergraphs, achieving near-optimal size and efficient update times, and extends it to parallel batch processing.

Contribution

It presents the first spectral sparsification algorithm for directed hypergraphs that works dynamically and in parallel batch settings.

Findings

Achieves near-optimal sparsifier size of O(n^2 / ε^2 log^7 m)

Provides amortized update time of O(r^2 log^3 m)

Extends to parallel batch-dynamic setting with efficient processing

Abstract

There has been a surge of interest in spectral hypergraph sparsification, a natural generalization of spectral sparsification for graphs. In this paper, we present a simple fully dynamic algorithm for maintaining spectral hypergraph sparsifiers of \textit{directed} hypergraphs. Our algorithm achieves a near-optimal size of $O(n^2 / \varepsilon ^2 \log ^7 m)$ and amortized update time of $O(r^2 \log ^3 m)$, where $n$ is the number of vertices, and $m$ and $r$ respectively upper bound the number of hyperedges and the rank of the hypergraph at any time. We also extend our approach to the parallel batch-dynamic setting, where a batch of any $k$ hyperedge insertions or deletions can be processed with $O(kr^2 \log ^3 m)$ amortized work and $O(\log ^2 m)$ depth. This constitutes the first spectral-based sparsification algorithm in this setting.