Document Open Access Logo

Listing 4-Cycles

Authors Amir Abboud, Seri Khoury, Oree Leibowitz, Ron Safier

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2023.25.pdf
  • Filesize: 0.78 MB
  • 16 pages

Document Identifiers

Author Details

Amir Abboud
  • Weizmann Institute of Science, Rehovot, Israel
Seri Khoury
  • UC Berkeley, CA, USA
Oree Leibowitz
  • Weizmann Institute of Science, Rehovot, Israel
Ron Safier
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Abboud, Amir: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No 101078482). Additionally, Amir Abboud is supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.

Cite As Get BibTex

Amir Abboud, Seri Khoury, Oree Leibowitz, and Ron Safier. Listing 4-Cycles. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.25

Abstract

We study the fine-grained complexity of listing all 4-cycles in a graph on n nodes, m edges, and t such 4-cycles. The main result is an Õ(min(n²,m^{4/3})+t) upper bound, which is best-possible up to log factors unless the long-standing O(min(n²,m^{4/3})) upper bound for detecting a 4-cycle can be broken. Moreover, it almost-matches recent 3-SUM-based lower bounds for the problem by Abboud, Bringmann, and Fischer (STOC 2023) and independently by Jin and Xu (STOC 2023). Notably, our result separates 4-cycle listing from the closely related triangle listing for which higher conditional lower bounds exist, and rule out such a "detection plus t" bound. We also show by simple arguments that our bound cannot be extended to mild generalizations of the problem such as reporting all pairs of nodes that participate in a 4-cycle.
[Independent work: Jin and Xu [Ce Jin and Yinzhan Xu, 2023] also present an algorithm with the same time bound.]

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Graph algorithms
  • cycles listing
  • subgraph detection
  • fine-grained complexity

Metrics

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

References

  1. Amir Abboud, Karl Bringmann, and Nick Fischer. Stronger 3-sum lower bounds for approximate distance oracles via additive combinatorics. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 391-404. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585240.
  2. Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1487-1500. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  3. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 522-539. SIAM, 2021. Google Scholar
  4. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. Google Scholar
  5. Guillaume Bagan, Arnaud Durand, and Etienne Grandjean. On acyclic conjunctive queries and constant delay enumeration. In International Workshop on Computer Science Logic, pages 208-222. Springer, 2007. Google Scholar
  6. R. Bar-Yehuda and S. Even. On approximating a vertex cover for planar graphs. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC '82, pages 303-309, New York, NY, USA, 1982. Association for Computing Machinery. URL: https://doi.org/10.1145/800070.802205.
  7. Jonathan Berry, Bruce Hendrickson, Randall LaViolette, and Cynthia Phillips. Tolerating the community detection resolution limit with edge weighting. Physical review. E, Statistical, nonlinear, and soft matter physics, 83:056119, May 2011. URL: https://doi.org/10.1103/PhysRevE.83.056119.
  8. Etienne Birmelé, Rui Ferreira, Roberto Grossi, Andrea Marino, Nadia Pisanti, Romeo Rizzi, and Gustavo Sacomoto. Optimal listing of cycles and st-paths in undirected graphs. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 1884-1896. SIAM, 2013. Google Scholar
  9. Andreas Björklund, Rasmus Pagh, Virginia Vassilevska Williams, and Uri Zwick. Listing triangles. In International Colloquium on Automata, Languages, and Programming, pages 223-234. Springer, 2014. Google Scholar
  10. Nofar Carmeli and Markus Kröll. On the enumeration complexity of unions of conjunctive queries. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 134-148, 2019. Google Scholar
  11. Nofar Carmeli and Markus Kröll. Enumeration complexity of conjunctive queries with functional dependencies. Theory of Computing Systems, 64(5):828-860, 2020. Google Scholar
  12. Nofar Carmeli and Luc Segoufin. Conjunctive queries with self-joins, towards a fine-grained enumeration complexity analysis. In Proceedings of the 42nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 277-289, 2023. Google Scholar
  13. Keren Censor-Hillel. Distributed subgraph finding: progress and challenges. arXiv preprint, 2022. URL: https://arxiv.org/abs/2203.06597.
  14. Yi-Jun Chang, Seth Pettie, and Hengjie Zhang. Distributed triangle detection via expander decomposition. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 821-840. SIAM, 2019. Google Scholar
  15. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  16. Søren Dahlgaard, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Finding even cycles faster via capped k-walks. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 112-120, 2017. Google Scholar
  17. Ran Duan, Hongxun Wu, and Renfei Zhou. Faster matrix multiplication via asymmetric hashing, 2022. URL: https://doi.org/10.48550/arXiv.2210.10173.
  18. Arnaud Durand and Etienne Grandjean. First-order queries on structures of bounded degree are computable with constant delay. ACM Transactions on Computational Logic (TOCL), 8(4):21-es, 2007. Google Scholar
  19. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science, 326(1-3):57-67, 2004. Google Scholar
  20. Fernando Florenzano, Cristian Riveros, Martín Ugarte, Stijn Vansummeren, and Domagoj Vrgoc. Constant delay algorithms for regular document spanners. In Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 165-177, 2018. Google Scholar
  21. Brooke Foucault Welles, Anne Van Devender, and Noshir Contractor. Is a friend a friend? investigating the structure of friendship networks in virtual worlds. In CHI '10 Extended Abstracts on Human Factors in Computing Systems, CHI EA '10, pages 4027-4032, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1753846.1754097.
  22. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. In Proceedings of the ninth annual ACM symposium on Theory of computing, pages 1-10, 1977. Google Scholar
  23. Rajesh Jayaram and John Kallaugher. An optimal algorithm for triangle counting in the stream. arXiv preprint, 2021. URL: https://arxiv.org/abs/2105.01785.
  24. Madhav Jha, C. Seshadhri, and Ali Pinar. A space efficient streaming algorithm for triangle counting using the birthday paradox. In Inderjit S. Dhillon, Yehuda Koren, Rayid Ghani, Ted E. Senator, Paul Bradley, Rajesh Parekh, Jingrui He, Robert L. Grossman, and Ramasamy Uthurusamy, editors, The 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, Chicago, IL, USA, August 11-14, 2013, pages 589-597. ACM, 2013. URL: https://doi.org/10.1145/2487575.2487678.
  25. Tao Jiang and Liana Yepremyan. Supersaturation of even linear cycles in linear hypergraphs. Combinatorics, Probability and Computing, 29(5):698-721, 2020. URL: https://doi.org/10.1017/S0963548320000206.
  26. Ce Jin and Yinzhan Xu. Removing additive structure in 3sum-based reductions. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 405-418. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585157.
  27. Donald B Johnson. Finding all the elementary circuits of a directed graph. SIAM Journal on Computing, 4(1):77-84, 1975. Google Scholar
  28. Ton Kloks, Dieter Kratsch, and Haiko Müller. Finding and counting small induced subgraphs efficiently. Information Processing Letters, 74(3-4):115-121, 2000. Google Scholar
  29. Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In SODA, 2016. Google Scholar
  30. François Le Gall. Improved quantum algorithm for triangle finding via combinatorial arguments. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 216-225. IEEE, 2014. Google Scholar
  31. Frédéric Magniez, Miklos Santha, and Mario Szegedy. Quantum algorithms for the triangle problem. SIAM Journal on Computing, 37(2):413-424, 2007. Google Scholar
  32. Dániel Marx and Michał Pilipczuk. Everything you always wanted to know about the parameterized complexity of subgraph isomorphism (but were afraid to ask). arXiv preprint, 2013. URL: https://arxiv.org/abs/1307.2187.
  33. Robert Morris and David Saxton. The number of c2𝓁-free graphs. Advances in Mathematics, 298:534-580, 2016. URL: https://doi.org/10.1016/j.aim.2016.05.001.
  34. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 26(2):415-419, 1985. Google Scholar
  35. Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 603-610, 2010. Google Scholar
  36. Dana Richards and Arthur L. Liestman. Finding cycles of a given length. In B.R. Alspach and C.D. Godsil, editors, Annals of Discrete Mathematics (27): Cycles in Graphs, volume 115 of North-Holland Mathematics Studies, pages 249-255. North-Holland, 1985. URL: https://doi.org/10.1016/S0304-0208(08)73019-6.
  37. Luc Segoufin. Constant delay enumeration for conjunctive queries. ACM SIGMOD Record, 44(1):10-17, 2015. Google Scholar
  38. Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing, 6(3):505-517, 1977. URL: https://doi.org/10.1137/0206036.
  39. Virginia Vassilevska Williams, Joshua R Wang, Ryan Williams, and Huacheng Yu. Finding four-node subgraphs in triangle time. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms, pages 1671-1680. SIAM, 2014. Google Scholar
  40. Virginia Vassilevska Williams, Joshua R. Wang, Ryan Williams, and Huacheng Yu. Finding four-node subgraphs in triangle time. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 1671-1680, USA, 2015. Society for Industrial and Applied Mathematics. Google Scholar
  41. Virginia Vassilevska Williams and Yinzhan Xu. Monochromatic triangles, triangle listing and apsp. 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 786-797, 2020. Google Scholar
  42. Raphael Yuster and Uri Zwick. Finding even cycles even faster. SIAM Journal on Discrete Mathematics, 10(2):209-222, 1997. Google Scholar
  43. Raphael Yuster and Uri Zwick. Detecting short directed cycles using rectangular matrix multiplication and dynamic programming. In SODA, volume 4, pages 254-260, 2004. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact