Finding Most-Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time

Authors Kevin Hua, Daniel Li, Jaewoo Park, Thatchaphol Saranurak



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Kevin Hua
  • University of Michigan, Ann Arbor, MI, USA
Daniel Li
  • University of Michigan, Ann Arbor, MI, USA
Jaewoo Park
  • University of Michigan, Ann Arbor, MI, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA

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Kevin Hua, Daniel Li, Jaewoo Park, and Thatchaphol Saranurak. Finding Most-Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 87:1-87:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.87

Abstract

We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has been unimproved for more than two decades. Our work also removes an important bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length directed cycle in a graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain a near-linear time algorithm, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of using connectivity oracles subject to vertex failures to speed up a static graph algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graphs
  • Flows
  • Randomized Algorithms
  • Vertex Connectivity

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References

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