Partition Arguments in Multiparty Communication Complexity

TL;DR

This paper investigates the limitations of partition arguments in multiparty communication complexity, showing they can significantly underestimate true complexity in randomized and nondeterministic cases, and linking deterministic bounds to the log-rank conjecture.

Contribution

It demonstrates the exponential gap between partition argument bounds and actual complexity in randomized models and connects deterministic bounds to the log-rank conjecture.

Findings

Partition arguments can underestimate complexity exponentially in randomized models.

Existence of functions with large true complexity but small partition-based lower bounds.

Progress on the log-rank conjecture relates to bounds obtained via partition arguments.

Abstract

Consider the "Number in Hand" multiparty communication complexity model, where k players holding inputs x_1,...,x_k in {0,1}^n communicate to compute the value f(x_1,...,x_k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem. In this paper, we study the power of partition arguments. Our two main results are very different in nature: (i) For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Omega(n), while partition arguments can only yield an Omega(log n) lower bound. The same holds for nondeterministic communication complexity. (ii) For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the "log-rank conjecture" in communication complexity. We conclude with two results on the multiparty "fooling set technique", another method for obtaining communication complexity lower bounds.