Multiparty Communication Complexity and Threshold Circuit Size of AC^0

Abstract

We prove an n Ω(1) /4 k lower bound on the randomized k-party communication complexity of depth 4 AC 0 functions in the number-on-forehead (NOF) model for up to Θ(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC 0 function for ω(log log n) players. For non-constant k the bounds are larger than all previous lower bounds for any AC 0 function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC 0 by MAJ • SYMM • AND circuits, showing that the well-known quasipolynomial simulations of AC 0 by such circuits are qualitatively optimal, even for formulas of small constant depth. We also exhibit a depth 5 formula in NP cc k − BPP cc k for k up to Θ(log n) and derive an Ω(2 √ log n/ √ k ) lower bound on the randomized k-party NOF communication complexity of set disjointness for up to Θ(log 1/3 n) players which is significantly larger than the O(log log n) players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 AC 0 functions.

The NOF Multiparty Communication Complexity of Composed Functions

“…, 0). In a very recent paper, Sherstov [33] significantly improves on the bounds of [21], [10] and [6] . First we observe that this lower bound applies -via a simple reduction -to NOR • g when g's support size is 1.…”

“…The focus of previous research has been on proving lower bounds for composed functions by selecting a "hard" outside function and a convenient inside function (see e.g. [32,35,21,10,6, ?]). Our approach is to study composed functions without putting any restriction on g and obtain characterizations for the communication complexity of composed functions with respect to the choice of g. This dual approach is particularly interesting in the multiparty setting where the choice for g increases double exponentially in k.…”

Improved Separations between Nondeterministic and Randomized Multiparty Communication

“…Subsequent to our work, [BHN08a] extend our main results (Theorem 1.1 and Theorem 1.2) by proving the separation in Theorem 1.1 under the stronger requirement that the function f is computable by explicit (unbounded fan-in) circuits of depth 4 (albeit they can only handle (1 − Ω(1)) log n players, as opposed to (1 − δ ) log n for arbitrarily small δ in our results).…”

Improved Separations between Nondeterministic and Randomized Multiparty Communication

“…We also address the challenge of exhibiting functions computable by small (unbounded fan-in) constant-depth circuits that require high communication for k-player protocols, which is relevant to separating various circuit classes (see, e.g., [HG91,RW93,BHN08b]). Previous results [Cha07] give such functions for k < log log n. We offer a slight improvement and achieve k = A log log n for any (possibly large) constant A, where the depth of the circuit computing the function depends on A.…”