A strong direct product theorem for disjointness

Abstract

A strong direct product theorem states that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const• kn the success probability of solving k instances of size n can only be exponentially small in k. This solves an open problem of [KSW07, LSS08]. We also show that this bound even holds for $AM$-communication protocols with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov [Raz92].