A Composition Theorem via Conflict Complexity

TL;DR

This paper introduces conflict complexity as a new measure to analyze the composition of query complexities, establishing a tight bound for the randomized query complexity of composed functions.

Contribution

It presents a novel conflict complexity measure and proves a composition theorem relating it to randomized query complexity, providing new insights into complexity bounds.

Findings

Conflict complexity $oxed{ ext{chi}(g)}$ is $oxed{ ext{Omega}( ext{sqrt}( ext{R}(g)))}$.

The composition bound $oxed{ ext{R}_{1/3}(f igcirc g^n) = oxed{ ext{Omega}( ext{R}_{4/9}(f) imes ext{sqrt}( ext{R}_{1/3}(g)))}}$ is established.

The bound is shown to be optimal via an example from prior work.

Abstract

Let $\R(\cdot)$ stand for the bounded-error randomized query complexity. We show that for any relation $f \subseteq \{0,1\}^n \times \mathcal{S}$ and partial Boolean function $g \subseteq \{0,1\}^n \times \{0,1\}$, $\R_{1/3}(f \circ g^n) = \Omega(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$. Independently of us, Gavinsky, Lee and Santha \cite{newcomp} proved this result. By an example demonstrated in their work, this bound is optimal. We prove our result by introducing a novel complexity measure called the \emph{conflict complexity} of a partial Boolean function $g$, denoted by $\chi(g)$, which may be of independent interest. We show that $\chi(g) = \Omega(\sqrt{\R(g)})$ and $\R(f \circ g^n) = \Omega(\R(f) \cdot \chi(g))$.