An Optimal Algorithm for Certifying Monotone Functions
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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 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) ·log n) queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate in x^⋆ may require Ω(nC(f)) queries in the worst case.
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