Document Open Access Logo

Efficient Contractions of Dynamic Graphs - With Applications

Authors Monika Henzinger , Evangelos Kosinas , Robin Münk , Harald Räcke

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.36.pdf
  • Filesize: 0.89 MB
  • 14 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Graph Algorithms
  • Cut Sparsifiers
  • Dynamic Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A non-trivial minimum cut (NMC) sparsifier is a multigraph Ĝ that preserves all non-trivial minimum cuts of a given undirected graph G. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide an NMC sparsifier upon request at any point during the sequence of updates. We employ simple dynamic forest data structures to achieve a fast from-scratch construction of the sparsifier at query time. Based on the strength of the adversary and desired type of time bounds, the data structure comes with different guarantees. Specifically, let G be a fully dynamic simple graph with n vertices and minimum degree δ. Then our data structure supports an insertion/deletion of an edge to/from G in n^o(1) worst-case time. Furthermore, upon request, it can return w.h.p. an NMC sparsifier of G that has O(n/δ) vertices and O(n) edges, in Ô(n) time. The probabilistic guarantees hold against an adaptive adversary. Alternatively, the update and query times can be improved to Õ(1) and Õ(n) respectively, if amortized-time guarantees are sufficient, or if the adversary is oblivious. Throughout the paper, we use Õ to hide polylogarithmic factors and Ô to hide subpolynomial (i.e., n^o(1)) factors.
We discuss two applications of our new data structure. First, it can be used to efficiently report a cactus representation of all minimum cuts of a fully dynamic simple graph. Building this cactus for the NMC sparsifier instead of the original graph allows for a construction time that is sublinear in the number of edges. Against an adaptive adversary, we can with high probability output the cactus representation in worst-case Ô(n) time. Second, our data structure allows us to efficiently compute the maximal k-edge-connected subgraphs of undirected simple graphs, by repeatedly applying a minimum cut algorithm on the NMC sparsifier. Specifically, we can compute with high probability the maximal k-edge-connected subgraphs of a simple graph with n vertices and m edges in Õ(m+n²/k) time. This improves the best known time bounds for k = Ω(n^{1/8}) and naturally extends to the case of fully dynamic graphs.

Cite As Get BibTex

Monika Henzinger, Evangelos Kosinas, Robin Münk, and Harald Räcke. Efficient Contractions of Dynamic Graphs - With Applications. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.36

Author Details

Monika Henzinger
  • Institute of Science and Technology, Klosterneuburg, Austria
Evangelos Kosinas
  • Institute of Science and Technology, Klosterneuburg, Austria
Robin Münk
  • Technical University of Munich, Germany
Harald Räcke
  • Technical University of Munich, Germany

Funding

Monika Henzinger and Evangelos Kosinas: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MoDynStruct, No. 101019564) and the Austrian Science Fund (FWF) grant https://www.doi.org/10.55776/Z422 and grant https://www.doi.org/10.55776/I5982. Harald Räcke and Robin Münk: This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 498605858.

References

  1. Anders Aamand, Adam Karczmarz, Jakub Lacki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal decremental connectivity in non-sparse graphs. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 6:1-6:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.6.
  2. Joshua Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012. URL: https://doi.org/10.1137/090772873.
  3. András A. Benczúr and David R. Karger. Approximating s-t minimum cuts in Õ(n^2) time. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 47-55. ACM, 1996. URL: https://doi.org/10.1145/237814.237827.
  4. András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015. URL: https://doi.org/10.1137/070705970.
  5. Shiri Chechik, Thomas Dueholm Hansen, Giuseppe F. Italiano, Veronika Loitzenbauer, and Nikos Parotsidis. Faster algorithms for computing maximal 2-connected subgraphs in sparse directed graphs. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1900-1918. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.124.
  6. Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and Thatchaphol Saranurak. A deterministic algorithm for balanced cut with applications to dynamic connectivity, flows, and beyond. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1158-1167. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00111.
  7. Sebastian Forster, Danupon Nanongkai, Liu Yang, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2046-2065. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.126.
  8. Wai Shing Fung, Ramesh Hariharan, Nicholas J. A. Harvey, and Debmalya Panigrahi. A general framework for graph sparsification. SIAM J. Comput., 48(4):1196-1223, 2019. URL: https://doi.org/10.1137/16M1091666.
  9. Loukas Georgiadis, Giuseppe F. Italiano, Evangelos Kosinas, and Debasish Pattanayak. On maximal 3-edge-connected subgraphs of undirected graphs. CoRR, abs/2211.06521, 2022. URL: https://doi.org/10.48550/arXiv.2211.06521.
  10. Mohsen Ghaffari, Krzysztof Nowicki, and Mikkel Thorup. Faster algorithms for edge connectivity via random 2-out contractions. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1260-1279. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.77.
  11. Gramoz Goranci, Monika Henzinger, Danupon Nanongkai, Thatchaphol Saranurak, Mikkel Thorup, and Christian Wulff-Nilsen. Fully dynamic exact edge connectivity in sublinear time. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 70-86. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch3.
  12. Zhongtian He, Shang-En Huang, and Thatchaphol Saranurak. Cactus representation of minimum cuts: Derandomize and speed up. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 1503-1541. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.61.
  13. Monika Henzinger, Sebastian Krinninger, and Veronika Loitzenbauer. Finding 2-edge and 2-vertex strongly connected components in quadratic time. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 713-724. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_58.
  14. Monika Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edge connectivity. SIAM J. Comput., 49(1):1-36, 2020. URL: https://doi.org/10.1137/18M1180335.
  15. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, 2001. URL: https://doi.org/10.1145/502090.502095.
  16. Bruce M. Kapron, Valerie King, and Ben Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1131-1142. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.81.
  17. David R. Karger. Minimum cuts in near-linear time. J. ACM, 47(1):46-76, 2000. URL: https://doi.org/10.1145/331605.331608.
  18. David R. Karger and Debmalya Panigrahi. A near-linear time algorithm for constructing a cactus representation of minimum cuts. In Claire Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 246-255. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.28.
  19. Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time. J. ACM, 66(1):4:1-4:50, 2019. URL: https://doi.org/10.1145/3274663.
  20. Yin Tat Lee and He Sun. Constructing linear-sized spectral sparsification in almost-linear time. SIAM Journal on Computing, 47(6):2315-2336, 2018. URL: https://doi.org/10.1137/16M1061850.
  21. On-Hei Solomon Lo, Jens M. Schmidt, and Mikkel Thorup. Compact cactus representations of all non-trivial min-cuts. Discret. Appl. Math., 303:296-304, 2021. URL: https://doi.org/10.1016/J.DAM.2020.03.046.
  22. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL: https://doi.org/10.1007/BF01758778.
  23. Hiroshi Nagamochi and Toshihide Ibaraki. Algorithmic Aspects of Graph Connectivity, volume 123 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2008. URL: https://doi.org/10.1017/CBO9780511721649.
  24. Chaitanya Nalam and Thatchaphol Saranurak. Maximal k-edge-connected subgraphs in weighted graphs via local random contraction. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 183-211. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch8.
  25. Thatchaphol Saranurak and Wuwei Yuan. Maximal k-edge-connected subgraphs in almost-linear time for small k. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 92:1-92:9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.92.
  26. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913-1926, 2011. URL: https://doi.org/10.1137/080734029.
  27. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM Journal on Computing, 40(4):981-1025, 2011. URL: https://doi.org/10.1137/08074489X.
  28. Mikkel Thorup. Fully-dynamic min-cut. Comb., 27(1):91-127, 2007. URL: https://doi.org/10.1007/s00493-007-0045-2.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact