Listing 6-Cycles in Sparse Graphs

Authors Virginia Vassilevska Williams , Alek Westover



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Virginia Vassilevska Williams
  • MIT, Cambridge, MA, USA
Alek Westover
  • MIT, Cambridge, MS, USA

Acknowledgements

We'd like to thank Nathan S. Sheffield, Claire Zhang, and Ce Jin for helpful discussions.

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Virginia Vassilevska Williams and Alek Westover. Listing 6-Cycles in Sparse Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 92:1-92:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.92

Abstract

This work considers the problem of output-sensitive listing of occurrences of 2k-cycles for fixed constant k ≥ 2 in an undirected host graph with m edges and t 2k-cycles. Recent work of Jin and Xu (and independently Abboud, Khoury, Leibowitz, and Safier) [STOC 2023] gives an O(m^{4/3}+t) time algorithm for listing 4-cycles, and recent work by Jin, Vassilevska Williams and Zhou [SOSA 2024] gives an Õ(n²+t) time algorithm for listing 6-cycles in n node graphs. We focus on resolving the next natural question: obtaining listing algorithms for 6-cycles in the sparse setting, i.e., in terms of m rather than n. Previously, the best known result here is the better of Jin, Vassilevska Williams and Zhou’s Õ(n²+t) algorithm and Alon, Yuster and Zwick’s O(m^{5/3}+t) algorithm.
We give an algorithm for listing 6-cycles with running time Õ(m^{1.6}+t). Our algorithm is a natural extension of Dahlgaard, Knudsen and Stöckel’s [STOC 2017] algorithm for detecting a 2k-cycle. Our main technical contribution is the analysis of the algorithm which involves a type of "supersaturation" lemma relating the number of 2k-cycles in a bipartite graph to the sizes of the parts in the bipartition and the number of edges. We also give a simplified analysis of Dahlgaard, Knudsen and Stöckel’s 2k-cycle detection algorithm (with a small polylogarithmic increase in the running time), which is helpful in analyzing our listing algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph algorithms
  • cycles listing
  • fine-grained complexity
  • sparse graphs

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References

  1. Amir Abboud, Karl Bringmann, and Nick Fischer. Stronger 3-sum lower bounds for approximate distance oracles via additive combinatorics. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 391-404, 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 Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1487-1500, 2022. URL: https://doi.org/10.1145/3519935.3520066.
  3. Amir Abboud, Seri Khoury, Oree Leibowitz, and Ron Safier. Listing 4-cycles. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2211.10022.
  4. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  5. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: https://doi.org/10.1007/BF02523189.
  6. John A Bondy and Miklós Simonovits. Cycles of even length in graphs. Journal of Combinatorial Theory, Series B, 16(2):97-105, 1974. Google Scholar
  7. Karl Bringmann and Egor Gorbachev. A fine-grained classification of subquadratic patterns for subgraph listing and friends. CoRR, abs/2404.04369, 2024. URL: https://doi.org/10.48550/arXiv.2404.04369.
  8. Søren Dahlgaard, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Finding even cycles faster via capped k-walks, 2017. URL: https://arxiv.org/abs/1703.10380.
  9. Pierre-Louis Giscard, Paul Rochet, and Richard C Wilson. Evaluating balance on social networks from their simple cycles. Journal of Complex Networks, 5(5):750-775, 2017. URL: https://doi.org/10.1093/COMNET/CNX005.
  10. 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.
  11. Ce Jin, Virginia Vassilevska Williams, and Renfei Zhou. Listing 6-cycles. In Proceedings of the Symposium on Simplicity in Algorithms (SOSA 2024), pages 19-27, 2024. URL: https://doi.org/10.1137/1.9781611977936.3.
  12. Ce Jin and Yinzhan Xu. Removing additive structure in 3sum-based reductions. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 405-418, 2023. URL: https://doi.org/10.1145/3564246.3585157.
  13. Ce Jin and Renfei Zhou. Personal communication, 2024. Google Scholar
  14. Steffen Klamt and Axel von Kamp. Computing paths and cycles in biological interaction graphs. BMC bioinformatics, 10:1-11, 2009. URL: https://doi.org/10.1186/1471-2105-10-181.
  15. Andrea Lincoln and Nikhil Vyas. Algorithms and Lower Bounds for Cycles and Walks: Small Space and Sparse Graphs. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.11.
  16. V. Vassilevska. Efficient algorithms for path problems. Ph.D. Thesis in Computer Science, Carnegie Mellon University, 2008. Google Scholar
  17. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5), August 2018. URL: https://doi.org/10.1145/3186893.
  18. Alek Westover. Listing-c6s-lp. https://github.com/awestover/listing-C6s-LP/tree/main, 2024. Accessed: 2024-08-08.
  19. Raphael Yuster and Uri Zwick. Finding even cycles even faster. SIAM Journal on Discrete Mathematics, 10(2):209-222, 1997. URL: https://doi.org/10.1137/S0895480194274133.
  20. Yufei Zhao. Graph Theory and Additive Combinatorics: Exploring Structure and Randomness. Cambridge University Press, 2023. Google Scholar
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