Volume 14 (2018) Article 22 pp. 1-17
On Multiparty Communication with Large versus Unbounded Error
Revised: March 24, 2017
Published: December 28, 2018
The unbounded-error communication complexity of a Boolean function $F$ is the limit of the $\epsilon$-error randomized complexity of $F$ as $\epsilon\to1/2.$ Communication complexity with weakly unbounded error is defined similarly but with an additive penalty term that depends on $1/2-\epsilon$. Explicit functions are known whose two-party communication complexity with unbounded error is logarithmic compared to their complexity with weakly unbounded error. Chattopadhyay and Mande (ECCC TR16-095, Theory of Computing 2018) recently generalized this exponential separation to the number-on-the-forehead multiparty model. We show how to derive such an exponential separation from known two-party work, achieving a quantitative improvement along the way. We present several proofs here, some as short as half a page.
In more detail, we construct a $k$-party communication problem $F\colon(\{0,1\}^{n})^{k}\to\{0,1\}$ that has complexity $O(\log n)$ with unbounded error and $\Omega(\sqrt n\,/\,4^{k})$ with weakly unbounded error, reproducing the bounds of Chattopadhyay and Mande. In addition, we prove a quadratically stronger separation of $O(\log n)$ versus $\Omega(n\,/\,4^k)$ using a nonconstructive argument.