Document Open Access Logo

Witness-Sensitive Detection of Induced Diamonds

Authors Keren Censor-Hillel , Tomer Even , Virginia Vassilevska Williams , Nathan Wallheimer

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.52.pdf
  • Filesize: 1.45 MB
  • 22 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Induced diamond detection
  • Witness-sensitive algorithms
  • Matrix multiplication
  • Subgraph detection
  • Fine-grained complexity

Metrics

Abstract

We provide a fast witness-sensitive algorithm for detecting an induced diamond (a K₄ minus an edge) in an n-vertex graph containing t induced diamonds. Our algorithm runs in time Õ(min(n^2.425/t^0.25 + n², n^ω)) with high probability, improving upon the prior state of the art (witness-oblivious) algorithm that runs in time O(n^ω log n) [Vassilevska Williams, Wang, Williams, Yu, SODA 2014] whenever t ≥ n^{(3-ω)/3}, where ω < 2.372 is the matrix multiplication exponent.
Our key insight is that the size of a clique containing one of the triangles of an induced diamond plays a crucial role in detecting such a diamond. We say that a diamond is r-heavy if this size is at least r, and we provide a fast detection algorithm for r-heavy diamonds in Õ(r⋅(n/r)^ω + (n/r)³+ nr) time. When there are no r-heavy diamonds, we provide a different fast detection algorithm in Õ(MM(n,n,n√{r/t})) time, where MM(a,b,c) denotes the time to multiply an a × b matrix by a b × c matrix, which is conditionally optimal for r = Õ(1).
Our main technical contribution is in designing a refinement framework for sampling vectors, which allows sampling vertices for detecting diamonds in a manner that is adaptive to the structure of graphs with no r-heavy diamonds. We establish that our technique is of a wide applicability, by showing how it also allows for faster witness-sensitive algorithms for 4-SUM and for a special case of 4-cycles.

Cite As Get BibTex

Keren Censor-Hillel, Tomer Even, Virginia Vassilevska Williams, and Nathan Wallheimer. Witness-Sensitive Detection of Induced Diamonds. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 52:1-52:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.52

Author Details

Keren Censor-Hillel
  • Department of Computer Science, Technion, Haifa, Israel
Tomer Even
  • Department of Computer Science, Technion, Haifa, Israel
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Nathan Wallheimer
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Censor-Hillel, Keren: The research is supported in part by the Israel Science Foundation (grant 529/23).
  • Vassilevska Williams, Virginia: Supported by NSF Grant CCF-2330048, BSF Grant 2024233, and a Simons Investigator Award.

References

  1. Amir Abboud, Shyan Akmal, and Nick Fischer. A truly subcubic combinatorial algorithm for induced 4-cycle detection. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3562-3599. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.130.
  2. Amir Abboud, Karl Bringmann, Nick Fischer, and Marvin Künnemann. The time complexity of fully sparse matrix multiplication. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4670-4703. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.167.
  3. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI symposium on Principles of Database Systems, pages 5-14, 2012. URL: https://doi.org/10.1145/2213556.2213560.
  4. Maryam Aliakbarpour, Amartya Shankha Biswas, Themis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee. Sublinear-time algorithms for counting star subgraphs via edge sampling. Algorithmica, 80:668-697, 2018. URL: https://doi.org/10.1007/S00453-017-0287-3.
  5. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Proceedings of the 2025 ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, page to appear. SIAM, 2025. Google Scholar
  6. N. Alon, Phuong Dao, I. Hajirasouliha, F. Hormozdiari, and S. C. Sahinalp. Biomolecular network motif counting and discovery by color coding. Bioinformatics, 24:i241-i249, 2008. Google Scholar
  7. N. Alon and J. Fox. Easily testable graph properties. Combinatorics, Probability and Computing, 24:646-657, 2015. URL: https://doi.org/10.1017/S0963548314000765.
  8. Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000. URL: https://doi.org/10.1007/S004930070001.
  9. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM (JACM), 42(4):844-856, 1995. URL: https://doi.org/10.1145/210332.210337.
  10. Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna. A simple sublinear-time algorithm for counting arbitrary subgraphs via edge sampling. arXiv preprint arXiv:1811.07780, 2018. URL: https://arxiv.org/abs/1811.07780.
  11. Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna. A simple sublinear-time algorithm for counting arbitrary subgraphs via edge sampling. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 6:1-6:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.6.
  12. Suman K. Bera and Amit Chakrabarti. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.STACS.2017.11.
  13. Amartya Shankha Biswas, T. Eden, and R. Rubinfeld. Towards a decomposition-optimal algorithm for counting and sampling arbitrary motifs in sublinear time. ArXiv, abs/2107.06582, 2021. URL: https://arxiv.org/abs/2107.06582.
  14. Antoine Bordes, Sumit Chopra, and Jason Weston. Question answering with subgraph embeddings. In Alessandro Moschitti, Bo Pang, and Walter Daelemans, editors, Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing, EMNLP 2014, October 25-29, 2014, Doha, Qatar, A meeting of SIGDAT, a Special Interest Group of the ACL, pages 615-620. ACL, 2014. URL: https://doi.org/10.3115/V1/D14-1067.
  15. Giorgos Bouritsas, Fabrizio Frasca, S. Zafeiriou, and M. Bronstein. Improving graph neural network expressivity via subgraph isomorphism counting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45:657-668, 2020. Google Scholar
  16. Keren Censor-Hillel. Distributed subgraph finding: progress and challenges. arXiv preprint arXiv:2203.06597, 2022. URL: https://doi.org/10.48550/arXiv.2203.06597.
  17. Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Fast approximate counting of cycles. In 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 37:1-37:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.37.
  18. Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Output-sensitive approximate counting via a measure-bounded hyperedge oracle, or: How asymmetry helps estimate k-clique counts faster. CoRR, abs/2503.21655, 2025. URL: https://doi.org/10.48550/arXiv.2503.21655.
  19. Nina Chiarelli, Berenice Martínez-Barona, Martin Milanič, Jérôme Monnot, and Peter Muršič. Strong cliques in diamond-free graphs. Theoretical Computer Science, 858:49-63, 2021. URL: https://doi.org/10.1016/J.TCS.2020.12.001.
  20. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  21. Mina Dalirrooyfard, Thuy Duong Vuong, and Virginia Vassilevska Williams. Graph pattern detection: Hardness for all induced patterns and faster non-induced cycles. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1167-1178, 2019. URL: https://doi.org/10.1145/3313276.3316329.
  22. Mina Dalirrooyfard and V. V. Williams. Induced cycles and paths are harder than you think. 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 531-542, 2022. Google Scholar
  23. Phuong Dao, Kendric Wang, Colin Collins, Martin Ester, Anna Lapuk, and S Cenk Sahinalp. Optimally discriminative subnetwork markers predict response to chemotherapy. Bioinformatics, 27(13):i205-i213, 2011. Google Scholar
  24. Holger Dell, John Lapinskas, and Kitty Meeks. Approximately counting and sampling small witnesses using a colorful decision oracle. SIAM J. Comput., 51(4):849-899, 2022. URL: https://doi.org/10.1137/19M130604X.
  25. Benjamin Doerr and Frank Neumann. Theory of Evolutionary Computation: Recent Developments in Discrete Optimization. Springer Nature, 2019. Google Scholar
  26. Devdatt P Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009. Google Scholar
  27. Talya Eden, Amit Levi, Dana Ron, and C Seshadhri. Approximately counting triangles in sublinear time. SIAM Journal on Computing, 46(5):1603-1646, 2017. URL: https://doi.org/10.1137/15M1054389.
  28. Talya Eden, Reut Levi, Dana Ron, and Ronitt Rubinfeld. Approximately counting and sampling hamiltonian motifs in sublinear time. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 1043-1054, 2025. URL: https://doi.org/10.1145/3717823.3718160.
  29. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science, 326(1-3):57-67, 2004. URL: https://doi.org/10.1016/J.TCS.2004.05.009.
  30. Hendrik Fichtenberger, Mingze Gao, and Pan Peng. Sampling arbitrary subgraphs exactly uniformly in sublinear time. ArXiv, abs/2005.01861, 2020. URL: https://arxiv.org/abs/2005.01861.
  31. Lior Gishboliner and A. Shapira. Deterministic vs non-deterministic graph property testing. Israel Journal of Mathematics, 204:397-416, 2013. Google Scholar
  32. Goldreich and Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, February 2002. URL: https://doi.org/10.1007/S00453-001-0078-7.
  33. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998. URL: https://doi.org/10.1145/285055.285060.
  34. Xiaohan Huang and Victor Y Pan. Fast rectangular matrix multiplication and applications. Journal of complexity, 14(2):257-299, 1998. URL: https://doi.org/10.1006/JCOM.1998.0476.
  35. 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
  36. John Kallaugher, Andrew McGregor, Eric Price, and Sofya Vorotnikova. The complexity of counting cycles in the adjacency list streaming model. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 119-133, 2019. URL: https://doi.org/10.1145/3294052.3319706.
  37. Tali Kaufman, Michael Krivelevich, and Dana Ron. Tight bounds for testing bipartiteness in general graphs. SIAM Journal on computing, 33(6):1441-1483, 2004. URL: https://doi.org/10.1137/S0097539703436424.
  38. Ton Kloks, Dieter Kratsch, and Haiko Müller. Finding and counting small induced subgraphs efficiently. Information Processing Letters, 74(3-4):115-121, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00047-8.
  39. Michihiro Kuramochi and G. Karypis. Frequent subgraph discovery. Proceedings 2001 IEEE International Conference on Data Mining, pages 313-320, 2001. Google Scholar
  40. François Le Gall and Masayuki Miyamoto. Lower Bounds for Induced Cycle Detection in Distributed Computing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021), volume 212 of Leibniz International Proceedings in Informatics (LIPIcs), pages 58:1-58:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.58.
  41. Xin Liu, Haojie Pan, Mutian He, Yangqiu Song, and Xin Jiang. Neural subgraph isomorphism counting. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2019. Google Scholar
  42. Dániel Marx. Can you beat treewidth? In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, pages 169-179. IEEE Computer Society, 2010. Google Scholar
  43. Andrew McGregor, Sofya Vorotnikova, and Hoa T Vu. Better algorithms for counting triangles in data streams. In Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 401-411, 2016. URL: https://doi.org/10.1145/2902251.2902283.
  44. Masayuki Miyamoto. Distributed complexity of p_k-freeness: Decision and certification. In Ho-Lin Chen, Wing-Kai Hon, and Meng-Tsung Tsai, editors, 36th International Symposium on Algorithms and Computation, ISAAC 2025, Tainan, Taiwan, December 7-10, 2025, 2025. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2025.51.
  45. Assaf Naor and Jacques Verstraëte. A note on bipartite graphs without 2k-cycles. Combinatorics, Probability and Computing, 14(5-6):845-849, 2005. Google Scholar
  46. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Comment. Math. Univ. Carol., 26(2):415-419, 1985. Google Scholar
  47. Stefan Neuwirth. The size of bipartite graphs with girth eight. arXiv preprint, 2001. URL: https://arxiv.org/abs/math/0102210.
  48. Amir Nikabadi and Janne Korhonen. Beyond distributed subgraph detection: Induced subgraphs, multicolored problems and graph parameters. In 25th International Conference on Principles of Distributed Systems, volume 217, 2022. Google Scholar
  49. Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Structures & Algorithms, 20(2):165-183, 2002. URL: https://doi.org/10.1002/RSA.10013.
  50. Syed Asad Rahman, Matthew Bashton, Gemma L Holliday, Rainer Schrader, and Janet M Thornton. Small molecule subgraph detector (smsd) toolkit. Journal of cheminformatics, 1:1-13, 2009. Google Scholar
  51. Qingyun Sun, Jianxin Li, Hao Peng, Jia Wu, Yuanxing Ning, Philip S Yu, and Lifang He. Sugar: Subgraph neural network with reinforcement pooling and self-supervised mutual information mechanism. In Proceedings of the web conference 2021, pages 2081-2091, 2021. URL: https://doi.org/10.1145/3442381.3449822.
  52. Charalampos Tsourakakis. The k-clique densest subgraph problem. In Proceedings of the 24th international conference on world wide web, pages 1122-1132, 2015. URL: https://doi.org/10.1145/2736277.2741098.
  53. Jakub Tětek. Approximate Triangle Counting via Sampling and Fast Matrix Multiplication. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 107:1-107:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.107.
  54. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. URL: https://doi.org/10.1145/3186893.
  55. 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
  56. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM (JACM), 49(3):289-317, 2002. URL: https://doi.org/10.1145/567112.567114.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact