Better Boosting of Communication Oracles, or Not

TL;DR

This paper investigates when naive error boosting in two-party communication protocols can be improved, focusing on oracles computable by constant-cost protocols, with implications for equality testing and complexity class completeness.

Contribution

It identifies conditions under which naive boosting can be improved, notably for the Equality oracle, and provides new insights into the complexity of equality testing and oracle cost growth.

Findings

Naive boosting can be improved for the Equality oracle but not for the 1-Hamming Distance oracle.

Established a superlinear growth in cost for computing multiple copies of certain problems.

Provided a new proof that Equality is not complete for constant-cost randomized communication.

Abstract

Suppose we have a two-party communication protocol for $f$ which allows the parties to make queries to an oracle computing $g$; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving $g$. If $q$ queries are made, the standard technique requires that we boost the error of each subroutine down to $O(1/q)$, leading to communication complexity which grows as $q \log q$. For which oracles $g$ can this naive boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing $k$ independent copies grows superlinear in $k$, drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).