Partition Bound Is Quadratically Tight for Product Distributions

Abstract

Let f: {0,1}^n*{0,1}^n -> {0,1} be a 2-party function. For every product distribution mu on {0,1}^n*{0,1}^n, we show that 

CC^{mu}_{0.49}(f) = O(log(prt_{1/8}(f))*log(log(prt_{1/8}(f)))^2), 

where CC^{mu}_{epsilon}(f) is the distributional communication complexity of f with error at most epsilon under the distribution mu and prt_{1/8}(f) is the partition bound of f, as defined by Jain and Klauck [Proc. 25th CCC, 2010]. We also prove a similar bound in terms of IC_{1/8}(f), the information complexity of f, namely, 

CC^{mu}_{0.49}(f) = O((IC_{1/8}(f)*log(IC_{1/8}(f)))^2). 

The latter bound was recently and independently established by Kol [Proc. 48th STOC, 2016] using a different technique.

We show a similar result for query complexity under product distributions. Let g: {0,1}^n -> {0,1} be a function. For every bit-wise product distribution mu on {0,1}^n, we show that

QC^{mu}_{0.49}(g) = O((log(qprt_{1/8}(g))*log(log(qprt_{1/8}(g))))^2), 

where QC^{mu}_{epsilon}(g) is the distributional query complexity of f with error at most epsilon under the distribution mu and qprt_{1/8}(g) is the query partition bound of the function g.

Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for product distributions.