TR08-003 Authors: Troy Lee, Adi Shraibman

Publication: 18th January 2008 13:02

Downloads: 1297

Keywords:

We show that disjointness requires randomized communication

Omega(\frac{n^{1/2k}}{(k-1)2^{k-1}2^{2^{k-1}}})

in the general k-party number-on-the-forehead model of complexity.

The previous best lower bound was Omega(\frac{log n}{k-1}). By

results of Beame, Pitassi, and Segerlind, this implies

2^{n^{Omega(1)}} lower bounds on the size of tree-like Lovasz-Schrijver

proof systems needed to refute certain unsatisfiable CNFs, and

super-polynomial lower bounds on the size of any tree-like proof system

whose terms are degree-d polynomial inequalities for

d = log log n - O(log log log n).

To prove our bound, we develop a new technique for showing lower bounds in the number-on-the-forehead model which

is based on the norm induced by cylinder intersections. This bound

naturally extends the linear program bound for rank useful in the

two-party case to the case of more than two parties, where the

fundamental concept of monochromatic rectangles is replaced by

monochromatic cylinder intersections. Previously, the only general

method known for showing lower bounds in the unrestricted

number-on-the-forehead model was the discrepancy method, which can

only show bounds of size O(log n) for disjointness.

To analyze the bound given by our new technique for the disjointness

function, we extend an elegant framework developed by Sherstov in the two-party case which relates polynomial degree to communication

complexity. Using this framework we are able to obtain bounds for any

tensor of the form F(x_1,\ldots,x_k) = f(x_1 \wedge \ldots \wedge x_k)

where f is a function which only depends on the number of ones in the

input.