Separation of Unbounded-Error Models in Multi-Party Communication Complexity.

We construct a simple function that has small unbounded-error communication complexity in the k-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with subexponential advantage over random guessing has cost essentially Omega(root n/4(k)) bits. This separates these classes up to k <= delta log n players for any constant delta < 1/4, and gives the largest known separation by an explicit function in this regime of k. Our analysis is elementary and self-contained, inspired by the methods of Goldmann, Hastad, and Razborov (Computational Complexity, 1992). After initial publication of our work as a preprint (ECCC, 2016), Sherstov pointed out to us that an alternative proof of an Omega((n/4(k))(1/7)) separation is implicit in his prior work (SICOMP, 2016). Furthermore, based on his prior work (SICOMP, 2013 and SICOMP, 2016), Sherstov gave an alternative proof of our constructive Omega(root n/4(k)) separation and also produced a stronger non-constructive Omega(root n/4(k)) separation. These results are explained in Sherstov's preprint (ECCC, 2016) and in article 14/22 in Theory of Computing.