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Vertex Sparsifiers for Hyperedge Connectivity

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

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Author Details

Han Jiang
  • University of Michigan, Ann Arbor, MI, USA
Shang-En Huang
  • University of Michigan, Ann Arbor, MI, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Tian Zhang
  • University of Michigan, Ann Arbor, MI, USA

Acknowledgements

We thank Sorrachai Yingchareonthawornchai, Yang Liu, Yunbum Kook, and Richard Peng for the discussion that inspires the notion of the pruned auxiliary graph in this paper. We also thank anonymous reviewers for their valuable comments.

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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) https://doi.org/10.4230/LIPIcs.ESA.2022.70

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Vertex sparsifier
  • hypergraph
  • connectivity

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