Almost quadratic gap between partition complexity and query/communication complexity

TL;DR

This paper demonstrates nearly quadratic separations between key complexity measures in query and communication complexity, revealing fundamental gaps and near-optimal bounds in computational complexity theory.

Contribution

It establishes the first nearly quadratic separation between deterministic query complexity and subcube partition complexity, and between deterministic communication complexity and the logarithm of the partition number.

Findings

Existence of a Boolean function with nearly quadratic gap between D(f) and D^{sc}(f)

Existence of a communication task with nearly quadratic gap between D^{cc}(f) and log chi(f)

Both separations are nearly optimal, matching known upper bounds

Abstract

We show nearly quadratic separations between two pairs of complexity measures: 1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$; 2. As a consequence, we obtain that there is a communication task $f(x, y)$ such that $D^{cc}(f)=\Omega(\log^{2-o(1)}\chi(f))$ where $D^{cc}(f)$ is the deterministic 2-party communication complexity of $f$ (in the standard 2-party model of communication) and $\chi(f)$ is the partition number of $f$. Both of those separations are nearly optimal: it is well known that $D(f)=O((D^{sc}(f))^{2})$ and $D^{cc}(f)=O(\log^2\chi(f))$.