Refuting Approaches to the Log-Rank Conjecture for XOR Functions

Authors Hamed Hatami , Kaave Hosseini , Shachar Lovett , Anthony Ostuni



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.82.pdf
  • Filesize: 0.68 MB
  • 11 pages

Document Identifiers

Author Details

Hamed Hatami
  • School of Computer Science, McGill University, Montreal, Canada
Kaave Hosseini
  • Department of Computer Science, University of Rochester, NY, USA
Shachar Lovett
  • Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA, USA
Anthony Ostuni
  • Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA, USA

Acknowledgements

This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing. A.O. thanks Daniel M. Kane for a number of helpful discussions. We also thank anonymous reviewers for useful comments on earlier versions of this manuscript.

Cite AsGet BibTex

Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 82:1-82:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.82

Abstract

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang, Wong, Xie, and Zhang (FOCS'13), and the second is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Communication complexity
  • log-rank conjecture
  • XOR functions
  • additive structure

Metrics

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

References

  1. Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards stronger counterexamples to the log-approximate-rank conjecture. In Mikolaj Bojanczyk and Chandra Chekuri, editors, Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 213 of LIPIcs, pages 13:1-13:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2021.13.
  2. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. SIAM Journal on Computing, 47(1):208-217, 2018. URL: https://doi.org/10.1137/17M1136869.
  3. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Log-rank and lifting for AND-functions. In Samir Khuller and Virginia Vassilevska Williams, editors, Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC), pages 197-208. ACM, 2021. URL: https://doi.org/10.1145/3406325.3450999.
  4. László Lovász and Michael Saks. Communication complexity and combinatorial lattice theory. Journal of Computer and System Sciences, 47(2):322-349, 1993. URL: https://doi.org/10.1016/0022-0000(93)90035-U.
  5. Shachar Lovett. Communication is bounded by root of rank. Journal of the ACM (JACM), 63(1):1:1-1:9, 2016. URL: https://doi.org/10.1145/2724704.
  6. Nikhil S Mande and Swagato Sanyal. On parity decision trees for Fourier-sparse boolean functions. In Nitin Saxena and Sunil Simon, editors, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 182 of LIPIcs, pages 29:1-29:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2020.29.
  7. Kurt Mehlhorn and Erik M Schmidt. Las Vegas is better than determinism in VLSI and distributed computing. In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H. Landweber, editors, Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing (STOC), pages 330-337. ACM, 1982. URL: https://doi.org/10.1145/800070.802208.
  8. Ashley Montanaro and Tobias Osborne. On the communication complexity of XOR functions. CoRR, abs/0909.3392, 2009. URL: https://doi.org/10.48550/arXiv.0909.3392.
  9. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
  10. Ryan O'Donnell, John Wright, Yu Zhao, Xiaorui Sun, and Li-Yang Tan. A composition theorem for parity kill number. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 144-154. IEEE, IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.22.
  11. Anup Rao and Amir Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, 2020. Google Scholar
  12. Amir Shpilka, Avishay Tal, and Ben Lee Volk. On the structure of boolean functions with small spectral norm. In Moni Naor, editor, Proceedings of the 5th Conference on Innovations in Theoretical Computer Science (ITCS), pages 37-48. ACM, 2014. URL: https://doi.org/10.1145/2554797.2554803.
  13. Benny Sudakov and István Tomon. Matrix discrepancy and the log-rank conjecture. CoRR, abs/2311.18524, 2023. URL: https://doi.org/10.48550/arXiv.2311.18524.
  14. Hing Yin Tsang, Chung Hoi Wong, Ning Xie, and Shengyu Zhang. Fourier sparsity, spectral norm, and the log-rank conjecture. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 658-667. IEEE, IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.76.
  15. Zhiqiang Zhang and Yaoyun Shi. On the parity complexity measures of boolean functions. Theoretical Computer Science, 411(26-28):2612-2618, 2010. URL: https://doi.org/10.1016/j.tcs.2010.03.027.