Fully Dynamic Spectral Sparsification of Hypergraphs

TL;DR

This paper introduces the first fully dynamic algorithm for spectral hypergraph sparsification, efficiently maintaining a sparse spectral approximation of a hypergraph under edge updates, extending graph sparsification techniques to hypergraphs.

Contribution

It presents a novel fully dynamic algorithm for spectral hypergraph sparsification, adapting spanner-based methods to hypergraphs and achieving efficient update times.

Findings

Maintains spectral sparsifiers with size $nr^3$ in dynamic hypergraphs.

Achieves amortized update time of $r^4$ polylogarithmic factors.

Extends dynamic spectral sparsification from graphs to hypergraphs.

Abstract

Spectral hypergraph sparsification, a natural generalization of the well-studied spectral sparsification notion on graphs, has been the subject of intensive research in recent years. In this work, we consider spectral hypergraph sparsification in the dynamic setting, where the goal is to maintain a spectral sparsifier of an undirected, weighted hypergraph subject to a sequence of hyperedge insertions and deletions. For any $0 < \varepsilon \leq 1$, we give the first fully dynamic algorithm for maintaining an $ (1 \pm \varepsilon) $-spectral hypergraph sparsifier of size $ n r^3 \operatorname{poly}\left( \log n, \varepsilon ^{-1} \right) $ with amortized update time $ r^4 \operatorname{poly}\left( \log n, \varepsilon ^{-1} \right) $, where $n$ is the number of vertices of the underlying hypergraph and $r$ is an upper-bound on the rank of hyperedges. Our key contribution is to show that the spanner-based sparsification algorithm of Koutis and Xu (2016) admits a dynamic implementation in the hypergraph setting, thereby extending the dynamic spectral sparsification framework for ordinary graphs by Abraham et al. (2016).