Multiparty Communication Complexity of Disjointness

TL;DR

This paper establishes strong lower bounds on the multiparty communication complexity of the Disjointness problem, revealing new insights into complexity classes and proof systems in the multiparty setting.

Contribution

It extends Sherstov's technique to the multiparty model, providing the first super-polylogarithmic lower bounds for constant k and separating key complexity classes.

Findings

Lower bound of n^Omega(1) for constant k

Super-polylogarithmic bounds for k=o(loglog n)

Implications for proof complexity lower bounds

Abstract

We obtain a lower bound of n^Omega(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 NP^{CC}_k and BPP^{CC}_k for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind \cite{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 \cite{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 \cite{Cha07}. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.