Document Open Access Logo

On Generalized Corners and Matrix Multiplication

Author Kevin Pratt



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2024.89.pdf
  • Filesize: 0.77 MB
  • 17 pages

Document Identifiers

Author Details

Kevin Pratt
  • Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, NY, USA

Acknowledgements

I thank Ryan O'Donnell for many useful discussions about these problems. I also thank Chris Umans for making comments which motivated this paper early on, and in particular, which motivated Theorem 30. I thank the reviewers for suggestions which improved the exposition of this paper.

Cite AsGet BibTex

Kevin Pratt. On Generalized Corners and Matrix Multiplication. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.89

Abstract

Suppose that S ⊆ [n]² contains no three points of the form (x,y), (x,y+δ), (x+δ,y'), where δ ≠ 0. How big can S be? Trivially, n ≤ |S| ≤ n². Slight improvements on these bounds are obtained from Shkredov’s upper bound for the corners problem [Shkredov, 2006], which shows that |S| ≤ O(n²/(log log n)^c) for some small c > 0, and a construction due to Petrov [Fedor Petrov, 2023], which shows that |S| ≥ Ω(n log n/√{log log n}). Could it be that for all ε > 0, |S| ≤ O(n^{1+ε})? We show that if so, this would rule out obtaining ω = 2 using a large family of abelian groups in the group-theoretic framework of [Cohn and Umans, 2003; Cohn et al., 2005] (which is known to capture the best bounds on ω to date), for which no barriers are currently known. Furthermore, an upper bound of O(n^{4/3 - ε}) for any fixed ε > 0 would rule out a conjectured approach to obtain ω = 2 of [Cohn et al., 2005]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Algebraic computation
  • fast matrix multiplication
  • additive combinatorics

Metrics

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

References

  1. Miklós Ajtai and Endre Szemerédi. Sets of lattice points that form no squares. Stud. Sci. Math. Hungar, 9(1975):9-11, 1974. Google Scholar
  2. Josh Alman and Jarosław Błasiok. Matrix multiplication and number on the forehead communication. arXiv preprint, 2023. URL: https://arxiv.org/abs/2302.11476.
  3. Josh Alman and Virginia Vassilevska Williams. Limits on all known (and some unknown) approaches to matrix multiplication. SIAM Journal on Computing, pages FOCS18-285, 2021. Google Scholar
  4. Noga Alon and Asaf Shapira. Testing subgraphs in directed graphs. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 700-709, 2003. Google Scholar
  5. Noga Alon, Amir Shpilka, and Christopher Umans. On sunflowers and matrix multiplication. In 2012 IEEE 27th Conference on Computational Complexity, pages 214-223. IEEE, 2012. Google Scholar
  6. Felix Behrend. On sequences of numbers not divisible one by another. Journal of the London Mathematical Society, 1(1):42-44, 1935. Google Scholar
  7. Felix A Behrend. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences, 32(12):331-332, 1946. Google Scholar
  8. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Eric Naslund, William F. Sawin, and Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete Anal., pages Paper No. 3, 1-27, 2017. URL: https://doi.org/10.19086/da.1245.
  9. Ashok K Chandra, Merrick L Furst, and Richard J Lipton. Multi-party protocols. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 94-99, 1983. Google Scholar
  10. Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Barriers for fast matrix multiplication from irreversibility. arXiv preprint, 2018. URL: https://arxiv.org/abs/1812.06952.
  11. Henry Cohn, Robert Kleinberg, Balázs Szegedy, and Christopher Umans. Group-theoretic algorithms for matrix multiplication. In Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS 2005), pages 379-388. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/SFCS.2005.39.
  12. Henry Cohn and Christopher Umans. A group-theoretic approach to fast matrix multiplication. In Proceedings of the 44th Annual Symposium on Foundations of Computer Science (FOCS 2003), pages 438-449. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238217.
  13. Jordan S Ellenberg and Dion Gijswijt. On large subsets of with no three-term arithmetic progression. Annals of Mathematics, pages 339-343, 2017. Google Scholar
  14. P Erdös, A Sárközy, and E Szemerédi. On a theorem of behrend. Journal of the Australian Mathematical Society, 7(1):9-16, 1967. Google Scholar
  15. Jacob Fox. A new proof of the graph removal lemma. Annals of Mathematics, pages 561-579, 2011. Google Scholar
  16. Jacob Fox and LászlóMiklós Lovász. A tight bound for green’s arithmetic triangle removal lemma in vector spaces. In Proceedings of the twenty-eighth annual acm-siam symposium on discrete algorithms, pages 1612-1617. SIAM, 2017. Google Scholar
  17. Ben Green. A Szemerédi-type regularity lemma in abelian groups, with applications. Geometric & Functional Analysis GAFA, 15(2):340-376, 2005. Google Scholar
  18. Robert Kleinberg, Will Sawin, and David E Speyer. The growth rate of tri-colored sum-free sets. arXiv preprint, 2016. URL: https://arxiv.org/abs/1607.00047.
  19. Fedor Petrov. A variant of the corners problem. MathOverflow, 2023. URL: https://mathoverflow.net/q/451594 (version: 2023-07-27).
  20. Will Sawin. Bounds for matchings in nonabelian groups. Electron. J. Combin., 25(4):Paper No. 4.23, 1-21, 2018. Google Scholar
  21. Arnold Schönhage. Partial and total matrix multiplication. SIAM Journal on Computing, 10(3):434-455, 1981. Google Scholar
  22. Oriol Serra, Lluís Vena, et al. A combinatorial proof of the removal lemma for groups. Journal of Combinatorial Theory, Series A, 116(4):971-978, 2009. Google Scholar
  23. Ilya D Shkredov. On a generalization of Szemerédi’s theorem. Proceedings of the London Mathematical Society, 93(3):723-760, 2006. Google Scholar
  24. V Strassen. Relative bilinear complexity and matrix multiplication. Journal für die reine und angewandte Mathematik, 374:406-443, 1987. Google Scholar
  25. Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13(4):354-356, 1969. Google Scholar
  26. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega. arXiv preprint, 2023. URL: https://arxiv.org/abs/2307.07970.
  27. Yufei Zhao. Graph Theory and Additive Combinatorics: Exploring Structure and Randomness. Cambridge University Press, 2023. 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