Multiparty Communication Complexity of Disjointness

We obtain a lower bound of n 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(log logn), the bounds remain super-polylogarithmic i.e. (logn). The previous best lower bound for three players until recently was Ω(logn). Our bound separates the communication complexity classes NP k and BPP k for k = o(log logn). Furthermore, by the results of Beame, Pitassi and Segerlind [4], our bound implies proof size lower bounds for tree-like, degree k − 1 threshold systems and superpolynomial size lower bounds for Lovász-Schrijver proofs. To obtain our result, we further develop the “Generalized Discrepancy Method” recently suggested by Sherstov [16]. The other main components of the proof are the “Approximation/Orthogonality Principle” that also appears in [16] and techniques to estimate discrepancy under non-uniform distribution developed by Chattopadhyay [8]. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.