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DOI: 10.4230/LIPIcs.ESA.2022.70

Vertex Sparsifiers for Hyperedge Connectivity

Authors: Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

Recently, Chalermsook et al. {[}SODA'21{]} introduces a notion of vertex sparsifiers for c-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for c-edge st-connectivity {[}Jin and Sun FOCS'22{]}. We study a natural extension called vertex sparsifiers for c-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph G = (V,E) with n vertices and m hyperedges with k terminal vertices and a parameter c, there exists a hypergraph H containing only O(kc³) hyperedges that preserves all minimum cuts (up to value c) between all subset of terminals. This matches the best bound of O(kc³) edges for normal graphs by [Liu'20]. Moreover, H can be constructed in almost-linear O(p^{1+o(1)} + n(rclog n)^{O(rc)}log m) time where r = max_{e ∈ E}|e| is the rank of G and p = ∑_{e ∈ E}|e| is the total size of G, or in poly(m, n) time if we slightly relax the size to O(kc³log^{1.5}(kc)) hyperedges.

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Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang. Vertex Sparsifiers for Hyperedge Connectivity. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jiang_et_al:LIPIcs.ESA.2022.70,
  author =	{Jiang, Han and Huang, Shang-En and Saranurak, Thatchaphol and Zhang, Tian},
  title =	{{Vertex Sparsifiers for Hyperedge Connectivity}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{70:1--70:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.70},
  URN =		{urn:nbn:de:0030-drops-170081},
  doi =		{10.4230/LIPIcs.ESA.2022.70},
  annote =	{Keywords: Vertex sparsifier, hypergraph, connectivity}
}

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Vertex Sparsifiers for Hyperedge Connectivity

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