Under the auspices of the Computational Complexity Foundation (CCF)
We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient protocols in any of the aforementioned measures.
Furthermore, we show that the notion of zero-communication complexity is equivalent to an extension of the common discrepancy bound.
Linial et al. [Combinatorica, 2007] showed that the discrepancy of a sign matrix is lower-bounded by an inverse polynomial in the logarithm of the associated matrix. We show that if these results can be generalized to the extended discrepancy, this will imply the log-rank conjecture.