A Strong Direct Product Theorem for Disjointness

TL;DR

This paper proves a strong direct product theorem for the Disjointness problem in randomized communication complexity, showing that solving multiple instances simultaneously requires proportionally more resources, with success probability decreasing exponentially.

Contribution

It establishes a new strong direct product theorem for Disjointness, including for protocols with limited ambiguity, and derives implications for 3-player protocols and matrix multiplication tradeoffs.

Findings

Exponential decay in success probability for multiple instances

Lower bounds for 3-player NOF protocols

Optimal tradeoffs for Boolean matrix multiplication

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 $const\cdot kn$ the success probability of solving $k$ instances of size $n$ can only be exponentially small in $k$. We show that this bound even holds for $AM$ communication protocols with limited ambiguity. This also 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.