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ABSTRACT:We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity.As we show, our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement.Our results also shed some light on the question how much communication can be saved by using entanglement. It is known that entanglement can save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound on the saving in communication is tight almost always.

In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well.

We study the -rank of a real matrix A, defined for any > 0 as the minimum rank over matrices that approximate every entry of A to within an additive . This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the -rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive approximation schemes for Nash equilibria for 2-player games when the payoff matrices are positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an efficient algorithm for the following problem: given a convex body A and a symmetric convex body B, find a covering a A with translates of B.

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Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in randomized, quantum, and even weaklyunbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f ⊕ g) = Θ(disc(f )disc(g)). As a consequence we obtain a strong direct product theorem for distributional complexity, and direct sum theorems for worst-case complexity, for bounds shown by the discrepancy *

We show that disjointness requires randomized communication Ωin 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 Lovász-Schrijver proofs.

We show that disjointness requires randomized communication Ωin 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 Lovász-Schrijver proofs.