A Direct Product Theorem for Discrepancy

Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in the distributional, randomized, quantum, and even unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f g) = (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 method. Our results resolve an open problem of Shaltiel (2003) who showed a weaker product theorem for discrepancy with respect to the uniform distribution, discUk (fk) = O(discU (f))k/3. The main tool for our results is semidefinite programming, in particular a recent characterization of discrepancy in terms of a semidefinite programming quantity by Linial and Shraibman (2006).

• Type: Conference
• Year: 2008
• Where: COCO
• Authors: Troy Lee, Adi Shraibman, Robert Spalek
• Keywords: Algorithms, COCO 2008, Optimal Product Theorem, Product Theorem, Semidefinite Programming