An Optimal Algorithm for Certifying Monotone Functions

TL;DR

This paper presents an optimal algorithm for certifying the value of monotone functions with query complexity matching the theoretical lower bound, extending to real-valued functions and establishing hardness results.

Contribution

It introduces an optimal query-efficient algorithm for certifying monotone functions and extends it to real-valued functions, resolving an open problem from STOC 2022.

Findings

Algorithm makes $O(C(f) \, \log n)$ queries, matching the lower bound.

Extends certification to real-valued monotone functions with $2C(f)$ certificates.

Proves hardness of finding shortest certificates, requiring exponential queries in worst case.

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.