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.
Beame, Pitassi, Segerlind, and Wigderson @cite_26 devised a method based on a direct product theorem to show a @math bound on the complexity of three-party disjointness in a model stronger than one-way where the first player speaks once, and then the two remaining players interact arbitrarily.
@cite_26
1831717676 2015924804 2770686134 2091731906
A strong direct product theorem states that, in order to solve k instances of a problem, if we provide less than k times the resource required to compute one instance, then the probability of overall success is exponentially small in k. In this paper, we consider the model of two-way public-coin communication complexity and show a strong direct product theorem for all relations in terms of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower bound method in this model. Our result therefore uniformly implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound, for example Inner Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result also implies near optimal direct product results for several important functions and relations used to show exponential separations between classical and quantum communication complexity, for which near optimal lower bounds are provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008] and Klartag and Regev [2011]. In fact we are not aware of any relation for which it is known that the smooth rectangle bound does not provide an optimal lower bound. This lower bound subsumes many of the other lower bound methods, for example the rectangle bound (a.k.a the corruption bound), the smooth discrepancy bound (a.k.a the bound) which in turn subsumes the discrepancy bound, the subdistribution bound and the conditional min-entropy bound. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [2012], in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012]. We prove n (1) lower bounds on the multiparty communication complexity of AC 0 functions in the number-on-forehead (NOF) model for up to �(logn) players. These are the first lower bounds for any AC 0 function for !(loglogn) players. In particular we show that there are families of depth 3 read-once AC 0 formulas having k-player randomized multiparty NOF communication complexity n (1) 2 O(k) . We show similar lower bounds for depth 4 read-once AC 0 formulas that have nondeterministic communication complexity O(log 2 n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC 0 functions. As a consequence of the latter bound, we obtain an n (1 k) 2 O(k) lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to �( p logn) players which is significantly larger than the up to �(loglogn) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(loglogn) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC 0 . We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity, of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.Our paradigm leads to two main results: (1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k > 2, approximating the kth frequency moment in the data stream model requires essentially Ω(n1-2 k) space; this resolves a conjecture of (J. Comput. System Sci. 58(1) (1999) 137).(2) A lower bound for the Lp approximation problem in the general communication model: this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360-369). As a consequence, we show that for p > 2, approximating the Lp norm to within a factor of ne in the data stream model with constant number of passes requires Ω(n1-4e-2 p) space. We give simple proofs of refinements of the complexity results of Gilboa and Zemel (1989), and we derive additional results of this sort. Our constructions employ imitation games, which are two person games in which both players have the same sets of pure strategies and the second player wishes to play the same pure strategy as the first player.