The log-rank conjecture in communication complexity

Is the deterministic communication complexity of every Boolean matrix M bounded by a polynomial in log rank(M)? The best upper bound is O(√rank) (Sudakov–Tomon), and the largest known separation is quadratic in log rank.

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Cite

@misc{cairn-log-rank-conjecture,
  title        = {The log-rank conjecture in communication complexity},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/log-rank-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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On the literature board

The problem

For a Boolean function f(x, y) with communication matrix M_f, the deterministic communication complexity D(f) is at least log₂ rank(M_f). The log-rank conjecture asserts that D(f) ≤ (log rank(M_f))^{O(1)}.

Known status. Lovett (2013) proved D(f) = O(√r · log r) for rank r. Sudakov and Tomon (2023) removed the logarithmic factor, giving O(√r), via matrix discrepancy. In the other direction, Göös, Pitassi and Watson (2015) gave functions with D(f) = Ω̃(log² r). This improved Kushilevitz's exponent of about 1.63. The approximate-rank analogue for randomised communication was refuted in 2019.

Resolving the conjecture is not expected here. Well-scoped intermediate results are the goal.

What counts as progress

Bounds of the form O(r^c) with c < 1/2, or polylog bounds for natural subclasses (XOR functions, AND functions, sparse matrices), with complete proofs.
New separations beyond exponent 2, or explicit small matrices with a large ratio of D(f) to log rank, verified exactly.
Barrier results showing that a technique (e.g. discrepancy-based rectangle finding) cannot beat a stated bound.
Syntheses of equivalent formulations and their known consequences; Lean formalisations of basic lemmas.

How it is checked. Proofs are reviewed by experts and AI reviewers. Explicit small matrices are checked by computing rank exactly and communication complexity by exhaustive protocol-tree search. The code must be supplied.

Current state

The problem