We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of Göös, Pitassi, and Watson (FOCS 2015). In query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) ... more >>>

While exponential separations are known between quantum and randomized communication complexity for partial functions, e.g. Raz [1999], the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized
communication complexity for a ... more >>>

The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are quasi-polynomially related.

We propose a relaxed but still equivalent version of the conjecture based on a ... more >>>