Document Open Access Logo

An Improved Protocol for the Exactly-N Problem

Authors Nati Linial, Adi Shraibman

PDF
Thumbnail PDF

File

LIPIcs.CCC.2021.2.pdf
  • Filesize: 0.57 MB
  • 8 pages

Document Identifiers

Author Details

Nati Linial
  • Hebrew University of Jerusalem, Israel
Adi Shraibman
  • The Academic College of Tel-Aviv-Yaffo, Israel

Funding

  • Linial, Nati: Supported in part by Grant 659/18 "High dimensional combinatorics" of the Israel Science Foundation.

Acknowledgements

We thank Noga Alon, Aya Bernstine and Alex Samorodnitsky for insightful discussions.

Cite AsGet BibTex

Nati Linial and Adi Shraibman. An Improved Protocol for the Exactly-N Problem. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.2

Abstract

In the 3-players exactly-N problem the players need to decide whether x+y+z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N]×[N].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Communication complexity
  • Number-On-the-Forehead
  • Corner-free sets

Metrics

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

References

  1. A. Ada, A. Chattopadhyay, O. Fawzi, and P. Nguyen. The NOF multiparty communication complexity of composed functions. computational complexity, 24(3):645-694, 2015. Google Scholar
  2. M. Ajtai and E. Szemerédi. Sets of lattice points that form no squares. Stud. Sci. Math. Hungar, 9(1975):9-11, 1974. Google Scholar
  3. N. Alon and A. Shraibman. Number on the forehead protocols yielding dense ruzsa-szemerédi graphs and hypergraphs. Acta Mathematica Hungarica, 161(2):488-506, 2020. Google Scholar
  4. P. Beame, M. David, T. Pitassi, and P. Woelfel. Separating deterministic from randomized nof multiparty communication complexity. In Proceedings of the 34th International Colloquium On Automata, Languages and Programming, Lecture Notes in Computer Science. Springer-Verlag, 2007. Google Scholar
  5. P. Beame, T. Pitassi, and N. Segerlind. Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity. SIAM Journal on Computing, 37(3):845-869, 2006. Google Scholar
  6. F. 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
  7. R. Beigel, W. Gasarch, and J. Glenn. The multiparty communication complexity of Exact-T: Improved bounds and new problems. In International Symposium on Mathematical Foundations of Computer Science, pages 146-156. Springer, 2006. Google Scholar
  8. T. F. Bloom and O. Sisask. Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions. arXiv preprint, 2020. URL: http://arxiv.org/abs/2007.03528.
  9. A. Chandra, M. Furst, and R. Lipton. Multi-party protocols. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 94-99. ACM, 1983. Google Scholar
  10. M. Elkin. An improved construction of progression-free sets. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 886-905. Society for Industrial and Applied Mathematics, 2010. Google Scholar
  11. W. T. Gowers. Hypergraph regularity and the multidimensional szemerédi theorem. Annals of Mathematics, pages 897-946, 2007. Google Scholar
  12. J. Håstad and M. Goldmann. On the power of small-depth threshold circuits. Computational Complexity, 1:113-129, 1991. Google Scholar
  13. E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  14. N. Linial, T. Pitassi, and A. Shraibman. On the communication complexity of high-dimensional permutations. In 10th Innovations in Theoretical Computer Science Conference, ITCS San Diego, California, USA, volume 124, pages 54:1-54:20, 2019. Google Scholar
  15. L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975. Google Scholar
  16. B. Nagle, V. Rödl, and M. Schacht. The counting lemma for regular k-uniform hypergraphs. Random Structures & Algorithms, 28(2):113-179, 2006. Google Scholar
  17. V. Rödl and J. Skokan. Regularity lemma for k-uniform hypergraphs. Random Structures & Algorithms, 25(1):1-42, 2004. Google Scholar
  18. K. F. Roth. On certain sets of integers. Journal of the London Mathematical Society, 1(1):104-109, 1953. Google Scholar
  19. I. Ruzsa and E. Szemerédi. Triple systems with no six points carrying three triangles. Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai, 18:939-945, 1978. Google Scholar
  20. J. Solymosi. Note on a generalization of Roth’s theorem. Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pages 825-827, 2003. Google Scholar
  21. E. Szemerédi. On sets of integers containing no k elements in arithmetic progression. Acta Arith, 27(199-245):2, 1975. Google Scholar
  22. B. L. van der Waerden. Beweis einer Baudetschen Vermutung. Nieuw Arch. Wiskunde, 15:212-216, 1927. Google Scholar
  23. A. Yao. On ACC and threshold circuits. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 619-627. IEEE, 1990. 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 Imprint Privacy Contact