Disjointness is hard in the multi-party number on the forehead model

TL;DR

This paper establishes new lower bounds on randomized communication complexity in the multi-party number-on-the-forehead model, revealing fundamental computational limitations and implications for proof complexity.

Contribution

It provides the first superpolynomial lower bounds for randomized multi-party communication complexity in the number-on-the-forehead model, showing a separation from nondeterministic complexity.

Findings

Established Omega(n^{1/(k+1)}/2^{2^k}) lower bounds for k-party randomized communication.

Separated nondeterministic and randomized complexity for up to log log n players.

Derived subexponential lower bounds on proof sizes for certain unsatisfiable CNFs.

Abstract

We show that disjointness requires randomized communication Omega(n^{1/(k+1)}/2^{2^k}) in the general k-party number-on-the-forehead model of complexity. The previous best lower bound for k >= 3 was log(n)/(k-1). Our results give a separation between nondeterministic and randomized multiparty number-on-the-forehead communication complexity for up to k=log log n - O(log log log n) many players. Also by a reduction of Beame, Pitassi, and Segerlind, these results imply subexponential lower bounds on the size of proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovasz-Schrijver proofs.