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Refuting Approaches to the Log-Rank Conjecture for XOR Functions

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

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

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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.

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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

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

Funding

  • Lovett, Shachar: Research supported by NSF awards 1953928 and 2006443, and a Simons investigator award.
  • Ostuni, Anthony: Research supported by NSF award 2006443.

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.

References

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