Nearly optimal separations between communication (or query) complexity and partitions

TL;DR

This paper demonstrates nearly optimal quadratic separations between various complexity measures such as communication, query, and partition complexities, advancing understanding of their relationships and establishing new superlinear separations.

Contribution

It introduces nearly quadratic and superlinear separations between communication/query complexities and partition measures, improving prior bounds and utilizing the cheat sheet framework.

Findings

Nearly quadratic separation between deterministic communication complexity and partition number.

Nearly quadratic separation between query complexity and subcube partition complexity.

First superlinear separation between quantum query complexity and subcube partition complexity.

Abstract

We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of G\"o\"os, Pitassi, and Watson (FOCS 2015). In query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) query complexity and subcube partition complexity, which is also essentially optimal. We also establish a nearly power 1.5 separation between quantum query complexity and subcube partition complexity, the first superlinear separation between the two measures. Lastly, we show a quadratic separation between quantum query complexity and one-sided subcube partition complexity. Our query complexity separations use the recent cheat sheet framework of Aaronson, Ben-David, and the author. Our query functions are built up in stages by alternating function composition with the cheat sheet construction. The communication complexity separation follows from lifting the query separation to communication complexity.