2026-08-07T13:42:14Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18146 2024-03-06T11:01:08Z ddc:004 open_access

Nearly Tight Spectral Sparsification of Directed Hypergraphs Oko, Kazusato Sakaue, Shinsaku Tanigawa, Shin-ichi Spectral sparsification (Directed) hypergraphs Iterative sampling 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kazusato Oko and Shinsaku Sakaue and Shin-ichi Tanigawa 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.94 urn:nbn:de:0030-drops-181469 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.94 eng https://creativecommons.org/licenses/by/4.0/legalcode