An Optimal Algorithm for Certifying Monotone Functions

Abstract

Given query access to a monotone function $f:\{0,1{\}}^n\to \{0,1\}$ with certificate complexity $C(f)$ and an input $x^{\star }$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{\star }$ certifying the value of $f(x^{\star })$. Our algorithm makes $O(C(f)\cdot \log n)$ queries to $f$, which matches the information-theoretic lower bound for this problem and resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch, Lange, and Tan [BKLT22]. We extend this result to an algorithm that finds a size-$2C(f)$ certificate for a real-valued monotone function with $O(C(f)\cdot \log n)$ queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate in $x^{\star }$ may require $\Omega \left(\left(\genfrac{}{}{0ex}{}{n}{C(f)}\right)\right)$ queries in the worst case.