A strong direct product theorem for the tribes function via the smooth-rectangle bound

TL;DR

This paper establishes an optimal strong direct product theorem for the Tribes function's communication complexity by leveraging the smooth-rectangle bound, providing an alternative proof of known lower bounds.

Contribution

It offers a new proof of the (n) lower bound for Tribes using the smooth-rectangle bound, and extends this to a strong direct product result.

Findings

Optimal (n) lower bound for Tribes via smooth-rectangle bound

Smooth-rectangle bound is at least as strong as the communication complexity for Tribes

Strong direct product theorem for Tribes using smooth-rectangle bound

Abstract

The main result of this paper is an optimal strong direct product result for the two-party public-coin randomized communication complexity of the Tribes function. This is proved by providing an alternate proof of the optimal lower bound of \Omega(n) for the randomised communication complexity of the Tribes function using the so-called smooth-rectangle bound, introduced by Jain and Klauck [JK10]. The optimal \Omega(n) lower bound for Tribes was originally proved by Jayram, Kumar and Sivakumar [JKS03], using a more powerful lower bound technique, namely the information complexity bound. The information complexity bound is known to be at least as strong a lower bound method as the smooth-rectangle bound [KLL+12]. On the other hand, we are not aware of any function or relation for which the smooth-rectangle bound is (asymptotically) smaller than its public-coin randomized communication complexity. The optimal direct product for Tribes is obtained by combining our smooth-rectangle bound for tribes with the strong direct product result of Jain and Yao [JY12] in terms of smooth-rectangle bound.