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Nearly Tight Spectral Sparsification of Directed Hypergraphs

Authors Kazusato Oko, Shinsaku Sakaue, Shin-ichi Tanigawa

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Author Details

Kazusato Oko
  • Department of Mathematical Informatics, The University of Tokyo, Japan
  • Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan
Shinsaku Sakaue
  • Department of Mathematical Informatics, The University of Tokyo, Japan
Shin-ichi Tanigawa
  • Department of Mathematical Informatics, The University of Tokyo, Japan

Funding

This work was supported by JST PRESTO Grant Number JPMJPR2126, JST ERATO Grant Number JPMJER1903, and JSPS KAKENHI Grant Number 20H05961.

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Kazusato Oko, Shinsaku Sakaue, and Shin-ichi Tanigawa. Nearly Tight Spectral Sparsification of Directed Hypergraphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 94:1-94:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.94

Abstract

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 ε-spectral sparsifier of the optimal O^*(n) size, where n is the number of vertices and O^* suppresses the ε^{-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^*(n²)-size ε-spectral sparsifier for a weighted directed hypergraph. Our result is optimal up to the ε^{-1} and log n factors since there is a lower bound of Ω(n²) even for directed graphs. We also show the first non-trivial lower bound of Ω(n²/ε) 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spectral sparsification
  • (Directed) hypergraphs
  • Iterative sampling

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