Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm
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Spectral hypergraph sparsification, which is 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 recently obtained an algorithm for constructing an ε-spectral sparsifier of optimal O^*(n) size, where O^* suppresses the ε^-1 and log n factors, while the optimal sparsifier size has not been known for directed hypergraphs. In this paper, we present the first algorithm for constructing an ε-spectral sparsifier for a directed hypergraph with O^*(n^2) hyperarcs. This improves the previous bound by Kapralov, Krauthgamer, Tardos, and Yoshida (2021), and it is optimal up to the ε^-1 and log n factors since there is a lower bound of Ω(n^2) even for directed graphs. For general directed hypergraphs, we show the first non-trivial lower bound of Ω(n^2/ε). Our algorithm can be regarded as an extension of the spanner-based graph sparsification by Koutis and Xu (2016). To exhibit the power of the spanner-based approach, we also examine a natural extension of Koutis and Xu's algorithm to undirected hypergraphs. We show that it outputs an ε-spectral sparsifier of an undirected hypergraph with O^*(nr^3) hyperedges, where r is the maximum size of a hyperedge. Our analysis of the undirected case is based on that of Bansal, Svensson, and Trevisan (2019), and the bound matches that of the hypergraph sparsification algorithm by Bansal et al. We further show that our algorithm inherits advantages of the spanner-based sparsification in that it is fast, can be implemented in parallel, and can be converted to be fault-tolerant.
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