An Optimal Algorithm for Certifying Monotone Functions

Abstract: Given query access to a monotone function $f\colon{0,1}^n\to{0,1}$ 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 $\Omega\left(\binom{n}{C(f)}\right)$ queries in the worst case.