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.

翻译：谱超图稀疏化作为图谱稀疏化的自然推广，近年来引起了广泛关注。本文提出了一种简单的全动态算法，用于维护\textit{有向}超图的谱稀疏化子。该算法实现了近乎最优的规模$O(n^2 / \varepsilon ^2 \log ^7 m)$与摊还更新时间为$O(r^2 \log ^3 m)$，其中$n$为顶点数，$m$与$r$分别表示任意时刻超边数量的上界和超图的秩。我们还将该方法扩展到并行批处理动态场景，其中任意$k$条超边的插入或删除批处理可在$O(kr^2 \log ^3 m)$的摊还工作量和$O(\log ^2 m)$的深度下完成。这构成了该场景下首个基于谱的稀疏化算法。