Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR22-044 | 4th April 2022 06:54

#### An Optimal Algorithm for Certifying Monotone Functions

TR22-044
Authors: Meghal Gupta, Naren Manoj
Publication: 4th April 2022 07:08
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.