A Composition Theorem via Conflict Complexity

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 $g \subseteq {0,1}ⁿ \times {0,1}$, $$\R_{1/3}(f \circ gⁿ⁾ = \Omega(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$$. 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 $$\R(f \circ gⁿ⁾ = \Omega(\R(f) \cdot \chi(g))$$.