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

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

Thumbnail PDF


  • Filesize: 0.9 MB
  • 19 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

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)


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
  • Graphs
  • Flows
  • Randomized Algorithms
  • Vertex Connectivity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Michael Becker, W. Degenhardt, Jürgen Doenhardt, Stefan Hertel, Gerd Kaninke, W. Kerber, Kurt Mehlhorn, Stefan Näher, Hans Rohnert, and Thomas Winter. A probabilistic algorithm for vertex connectivity of graphs. Inf. Process. Lett., 15(3):135-136, 1982. Google Scholar
  2. Keren Censor-Hillel, Mohsen Ghaffari, and Fabian Kuhn. Distributed connectivity decomposition. In PODC, pages 156-165. ACM, 2014. Google Scholar
  3. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time, 2022. URL:
  4. Joseph Cheriyan and John H. Reif. Directed s-t numberings, rubber bands, and testing digraph k-vertex connectivity. Combinatorica, 14(4):435-451, 1994. Announced at SODA'92. Google Scholar
  5. Joseph Cheriyan and Ramakrishna Thurimella. Algorithms for parallel k-vertex connectivity and sparse certificates (extended abstract). In STOC, pages 391-401. ACM, 1991. Google Scholar
  6. Joseph Cheriyan and Ramakrishna Thurimella. Fast algorithms for k-shredders and k-node connectivity augmentation. Journal of Algorithms, 33(1):15-50, 1999. URL:
  7. Abdol-Hossein Esfahanian and S. Louis Hakimi. On computing the connectivities of graphs and digraphs. Networks, 14(2):355-366, 1984. Google Scholar
  8. Shimon Even. An algorithm for determining whether the connectivity of a graph is at least k. SIAM J. Comput., 4(3):393-396, 1975. Google Scholar
  9. Shimon Even and Robert Endre Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4(4):507-518, 1975. Google Scholar
  10. Sebastian Forster, Danupon Nanongkai, Thatchaphol Saranurak, Liu Yang, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2046-2065. SIAM, 2020. Google Scholar
  11. Harold N. Gabow. Using expander graphs to find vertex connectivity. J. ACM, 53(5):800-844, 2006. Announced at FOCS'00. Google Scholar
  12. Zvi Galil. Finding the vertex connectivity of graphs. SIAM J. Comput., 9(1):197-199, 1980. Google Scholar
  13. Monika Rauch Henzinger. A static 2-approximation algorithm for vertex connectivity and incremental approximation algorithms for edge and vertex connectivity. J. Algorithms, 24(1):194-220, 1997. Google Scholar
  14. Monika Rauch Henzinger, Satish Rao, and Harold N. Gabow. Computing vertex connectivity: New bounds from old techniques. J. Algorithms, 34(2):222-250, 2000. Announced at FOCS'96. Google Scholar
  15. Tibor Jordán. On the number of shredders. Journal of Graph Theory, 31(3):195-200, 1999. URL:<195::AID-JGT4>3.0.CO;2-E.
  16. D Kleitman. Methods for investigating connectivity of large graphs. IEEE Transactions on Circuit Theory, 16(2):232-233, 1969. Google Scholar
  17. Evangelos Kosinas. Connectivity Queries Under Vertex Failures: Not Optimal, but Practical. In 31st Annual European Symposium on Algorithms (ESA 2023), volume 274 of Leibniz International Proceedings in Informatics (LIPIcs), pages 75:1-75:13, 2023. URL:
  18. Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Vertex connectivity in poly-logarithmic max-flows. CoRR, abs/2104.00104, 2021. URL:
  19. Nathan Linial, László Lovász, and Avi Wigderson. Rubber bands, convex embeddings and graph connectivity. Combinatorica, 8(1):91-102, 1988. Announced at FOCS'86. Google Scholar
  20. Yaowei Long and Thatchaphol Saranurak. Near-optimal deterministic vertex-failure connectivity oracles. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1002-1010, 2022. URL:
  21. David W. Matula. Determining edge connectivity in o(nm). In FOCS, pages 249-251. IEEE Computer Society, 1987. Google Scholar
  22. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of ak-connected graph. Algorithmica, 7(1-6):583-596, 1992. Google Scholar
  23. Danupon Nanongkai, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Breaking quadratic time for small vertex connectivity and an approximation scheme. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 241-252, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  24. Seth Pettie and Longhui Yin. The structure of minimum vertex cuts. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 105:1-105:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  25. VD Podderyugin. An algorithm for finding the edge connectivity of graphs. Vopr. Kibern, 2:136, 1973. Google Scholar
  26. Thatchaphol Saranurak and Sorrachai Yingchareonthawornchai. Deterministic small vertex connectivity in almost linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 789-800. IEEE Computer Society, 2022. URL: