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Refuting approaches to the log-rank conjecture for XOR functions

Hamed Hatami, Kaave Hosseini, Shachar Lovett, Anthony Ostuni

TL;DR

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 by constructing two specific boolean functions tailored to each.

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 et al. (FOCS'13), and the second one 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.

Refuting approaches to the log-rank conjecture for XOR functions

TL;DR

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 by constructing two specific boolean functions tailored to each.

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 et al. (FOCS'13), and the second one 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.
Paper Structure (10 sections, 5 theorems, 24 equations)

This paper contains 10 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Folding
  3. Refuting a greedy approach
  4. Refuting a randomized approach
  5. Overview.
  6. Preliminaries
  7. One excellent folding direction
  8. Many good folding directions
  9. Conclusion
  10. Acknowledgements

Key Result

Theorem 1.3

For infinitely many $n$, there is a function $f:\mathbb{F}_2^n\to\{-1,1\}$ such that for $\mathcal{S} = \mathrm{supp}(\widehat{f})$, it holds for all distinct $\gamma_1, \gamma_2 \in \mathbb{F}_2^n$.

Theorems & Definitions (21)

  • Conjecture 1.1: Log-rank conjecture lovasz1993communication
  • Conjecture 1.2: XOR log-rank conjecture
  • Theorem 1.3: Informal version of \ref{['thm:preciseGreedy']}
  • Remark 1.4
  • Conjecture 1.5: mande2020parity
  • Theorem 1.6: Informal version of \ref{['thm:quantifiedMain']}
  • Theorem 3.1
  • Example 3.2: Addressing
  • Example 3.3: Subspace addressing
  • Lemma 3.4
  • ...and 11 more