Relative-error unateness testing

TL;DR

This paper introduces new algorithms and lower bounds for testing whether a Boolean function is unate, using relative-error property testing with query complexities depending on the number of satisfying assignments.

Contribution

It presents the first non-adaptive and adaptive algorithms for relative-error unateness testing, along with matching lower bounds, advancing the understanding of property testing complexity.

Findings

Non-adaptive algorithm with $ ilde{O}(rac{ ext{log}(N)}{ ext{epsilon}})$ complexity

Adaptive algorithm that does not require $N$ with similar complexity

Lower bounds that match the upper bounds for non-adaptive testing

Abstract

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary $f: \{0,1\}^n \to \{0,1\}$ for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the $n$ input variables. Our first result is a one-sided non-adaptive algorithm for this problem that makes $\tilde{O}(\log(N)/\epsilon)$ samples and queries, where $N=|f^{-1}(1)|$ is the number of satisfying assignments of the function that is being tested and the value of $N$ is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of $N$ and with high probability makes $\tilde{O}(\log(N)/\epsilon)$ samples and queries. We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of $N$ up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.