The Distinguishability of Product Distributions by Read-Once Branching Programs

We improve the main result of Brody and Verbin from FOCS 2010 on the power of constant-width branching programs to distinguish product distributions. Specifically, we show that a coin must have bias at least Ω(1/log(n)^ω-2) to be distinguishable from a fair coin by a width w, length n read-once branching program (for each constant w), which is a tight bound. Our result introduces new techniques, in particular a novel "interwoven hybrid" technique and a "program randomization" technique, both of which play crucial roles in our proof. Using the same techniques, we also succeed in giving tight upper bounds on the maximum influence of monotone functions computable by width w read-once branching programs.