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Tight zero-error lower bound in terms of deterministic complexity

Determine whether the public-coin zero-error randomized communication complexity of every Boolean matrix F satisfies a lower bound of the form Z(F) ≥ c · sqrt(D(F)) for some universal constant c > 0, improving the known Ω(D(F)^{1/4}) bound.

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Background

Zero-error public-coin randomized complexity (ZPP) is polynomially related to deterministic complexity: the current best lower bound is Ω(D(F){1/4}). It is conjectured that the optimal relationship might be quadratic-root, which would be tight up to constants.

Improving this bound would sharpen our understanding of how randomness helps in zero-error settings relative to deterministic protocols.

References

It is an open problem whether this bound can be improved to $\Omega(\sqrt{D(F)})$, which, if true, would be sharp.

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in After Theorem 4.1, Section 5 (Probabilistic Communication Models, zero-error subsection)