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.

翻译：图谱超图稀疏化旨在将著名的图谱稀疏化方法扩展到超图，近年来已被广泛研究。对于无向超图，Kapralov、Krauthgamer、Tardos 和 Yoshida（2022）证明了最优规模为 $O^*(n)$ 的 $\varepsilon$-谱稀疏化子，其中 $n$ 为顶点数，$O^*$ 表示隐藏了 $\varepsilon^{-1}$ 和 $\log n$ 因子。然而，对于有向超图，最优稀疏化子的规模此前尚未可知。我们的主要贡献是提出了首个算法，可为加权有向超图构造规模为 $O^*(n^2)$ 的 $\varepsilon$-谱稀疏化子。由于即使对于有向图也存在 $\Omega(n^2)$ 的下界，因此我们的结果在 $\varepsilon^{-1}$ 和 $\log n$ 因子意义下是最优的。我们还给出了对于一般有向超图的首个非平凡下界 $\Omega(n^2/\varepsilon)$。该算法的基本思想借鉴了 Koutis 和 Xu（2016）基于支撑子图的普通图稀疏化方法。其迭代采样方法在设计多种场景下的稀疏化算法时确实有效。为证明这一点，我们还提出了一个针对无向超图的类似迭代采样算法，该算法达到了最优规模界之一，支持并行实现，并可转化为容错版本。