A Composition Theorem via Conflict Complexity

Let $\mathbb{R}(\cdot )$ 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 $\chi (g)=\mathrm{\Omega }(\sqrt{\mathbb{R}(g)})$ and $\R(f \circ g^n) = \Omega(\R(f) \cdot \chi(g))$.