Refuting approaches to the log-rank conjecture for XOR functions
CoRR(2023)
摘要
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.
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