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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.94
URN: urn:nbn:de:0030-drops-181469
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18146/
Oko, Kazusato ;
Sakaue, Shinsaku ;
Tanigawa, Shin-ichi
Nearly Tight Spectral Sparsification of Directed Hypergraphs
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.
BibTeX - Entry
@InProceedings{oko_et_al:LIPIcs.ICALP.2023.94,
author = {Oko, Kazusato and Sakaue, Shinsaku and Tanigawa, Shin-ichi},
title = {{Nearly Tight Spectral Sparsification of Directed Hypergraphs}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {94:1--94:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18146},
URN = {urn:nbn:de:0030-drops-181469},
doi = {10.4230/LIPIcs.ICALP.2023.94},
annote = {Keywords: Spectral sparsification, (Directed) hypergraphs, Iterative sampling}
}
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Keywords: |
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Spectral sparsification, (Directed) hypergraphs, Iterative sampling |
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Collection: |
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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05.07.2023 |
