Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications

Abstract

We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck [Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, 2010, pp. 247--258] and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm-based methods (e.g., the $\gamma_2$ method) and rectangle-based methods (e.g., the rectangle/corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols, where the players can either output a value or abort. We prove, using a sampling protocol designed by Braverman and Weinstein [in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Comput. Sci. 7408, Springer, Heidelberg, 2012, pp. 459--470], the following compression lemma: given a protocol for a function $f$ with information complexity $I$, one can construct a zero-communication protocol that has nonabort probability at least $2^{-O(I)}$ and that computes $f$ correctly with high probability conditioned on not aborting. Then, we show how such a zero-communication protocol relates to the relaxed partition bound. We use our main theorem to resolve three of the open questions raised by Braverman [Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 505--524]. First, we show that the information complexity of the Vector in Subspace Problem [B. Klartag and O. Regev, Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, 2011, pp. 31--40] is $\Omega(n^{1/3})$, which, in turn, implies that there exists an exponential separation between quantum communication complexity and classical information complexity. Moreover, we provide an $\Omega(n)$ lower bound on the information complexity of the Gap Hamming Distance Problem.