Partition Arguments in Multiparty Communication Complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x_1,..,x_k in {0, 1}^n (respectively) communicate in order to compute the value f(x_1,...,x_k). 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 the partition arguments method. Our two main results are very different in nature: (i) For randomized communication complexity we show that partition arguments may be exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Ω(n) but partition arguments can only yield an Ω(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” of communication complexity.