A composition theorem for randomized query complexity via max conflict complexity

Abstract

Let $R_{\epsilon }(\cdot )$ stand for the bounded-error randomized query complexity with error $\epsilon >0$. For any relation $f\subseteq \{0,1{\}}^n\times S$ and partial Boolean function $g\subseteq \{0,1{\}}^m\times \{0,1\}$, we show that $R_{1/3}(f\circ g^n)\in \Omega (R_{4/9}(f)\cdot \sqrt{R_{1/3}(g)})$, where $f\circ g^n\subseteq (\{0,1{\}}^m{)}^n\times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar{\chi }(g)$ of a partial Boolean function $g$. We show $\bar{\chi }(g)\in \Omega (\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f\circ g^n)\in \Omega (R_{4/9}(f)\cdot \bar{\chi }(g))$; these two bounds imply our composition result. We further show that $\bar{\chi }(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari.

26 pages