An Optimal Algorithm for Certifying Monotone Functions

Meghal Gupta, Naren Sarayu Manoj

Abstract: Given query access to a monotone function f: {0,1}n → {0,1} with certificate complexity C(f) and an input x*, we design an algorithm that outputs a size-C (f) subset of x* certifying the value of f(x*). Our algorithm makes O (C(f) · log n) queries to f, which matches the information-theoretic lower bound for this problem and resolves the main open question posed in the STOC 2022 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) · logn) queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate for x* may require queries in the worst case.