Multiparty Communication Complexity of Disjointness
Abstract
We obtain a lower bound of nOmega(1) on the k-party randomized communication complexity of the Disjointness function in the `Number on the Forehead' model of multiparty communication when k is a constant. For k=o(loglog n), the bounds remain super-polylogarithmic i.e. (log n)omega(1). The previous best lower bound for three players until recently was Omega(log n). Our bound separates the communication complexity classes NPCCk and BPPCCk for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind BPS07, our bound implies proof size lower bounds for tree-like, degree k-1 threshold systems and superpolynomial size lower bounds for Lovasz-Schrijver proofs. Sherstov She07b recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obtain our results by extending his technique to the multi-party setting using ideas from Chattopadhyay Cha07. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.