Partition Bound Is Quadratically Tight for Product Distributions

Let f : {0, 1} × {0, 1} → {0, 1} be a 2-party function. For every product distribution μ on {0, 1} × {0, 1}, we show that CCμ0.49(f) = O (( log prt1/8(f) · log log prt1/8(f) )2) , where CCε (f) is the distributional communication complexity of f with error at most ε under the distribution μ and prt1/8(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 IC1/8(f), the information complexity of f , namely, CCμ0.49(f) = O (( IC1/8(f) · log IC1/8(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} → {0, 1} be a function. For every bit-wise product distribution μ on {0, 1}, we show that QCμ0.49(g) = O (( log qprt1/8(g) · log log qprt1/8(g) )2) , where QCε (g) is the distributional query complexity of f with error at most ε under the distribution μ and qprt1/8(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. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, G.2.1 Combinatorics