On multiparty communication with large versus unbounded error

The communication complexity of F with unbounded error is the limit of the ε-error randomized complexity of F as ε → 1/2. Communication complexity with weakly bounded error is defined similarly but with an additive penalty term that depends on 1/2− ε . Explicit functions are known whose two-party communication complexity with unbounded error is exponentially smaller than with weakly bounded error. Chattopadhyay and Mande (ECCC Report TR16-095) recently generalize this exponential separation to the number-on-the-forehead multiparty model, with a rather technical proof from first principles. We show how to derive such an exponential separation from known two-party work, achieving stronger parameters along the way. We present several proofs for this result, some as short as half a page. Our strongest separation is a k-party communication problem F : ({0,1}n)k → {0,1} that has complexity O(logn) with unbounded error and Ω(n/4k) with weakly bounded error. ∗The author was supported in part by NSF CAREER award CCF-1149018 and an Alfred P. Sloan Foundation Research Fellowship. ACM Classification: F.1.3, F.2.3 AMS Classification: 68Q17, 68Q15