Relative-error unateness testing
Authors:
Xi Chen,
Diptaksho Palit,
Kabir Peshawaria,
William Pires,
Rocco A. Servedio,
Yiding Zhang
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 eac…
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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 $\tilde{O}(\log(N)/ε)$ 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 $\tilde{O}(\log(N)/ε)$ 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.
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Submitted 24 October, 2025;
originally announced October 2025.
Computational Complexity in Property Testing
Authors:
Renato Ferreira Pinto Jr.,
Diptaksho Palit,
Sofya Raskhodnikova
Abstract:
We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity, relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query interplay and develop tools for proving time c…
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We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity, relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query interplay and develop tools for proving time complexity lower bounds. Our first contribution is a pair of time-query hierarchy theorems for property testing. For all suitable nondecreasing functions $q(n)$ and $t(n)$ with $t(n)\geq q(n)$, we construct properties with query complexity $\tildeΘ(q(n))$ and time complexity $\tildeΩ(t(n))$. Our weak hierarchy holds unconditionally, whereas the strong version-assuming the Strong Exponential Time Hypothesis-provides better control over the time complexity of the constructed properties.
We then turn to halfspaces in $\mathbb{R}^d$, a fundamental class in property testing and learning theory. We study the problem of approximating the distance from the input function to the nearest halfspace within additive error $ε$. For the distribution-free distance approximation problem, known algorithms achieve query complexity $O(d/ε^2)$, but take time $\tildeΘ(1/ε^d)$. We provide a fine-grained justification for this gap: assuming the $k$-SUM conjecture, any algorithm must have running time $Ω(1/ε^{d/2})$. This fine-grained lower bound yields a provable separation between query and time complexity for a natural and well-studied (tolerant) testing problem. We also prove that any Statistical Query (SQ) algorithm under the standard Gaussian distribution requires $(1/ε)^{Ω(d)}$ queries if the queries are answered with additive error up to $ε^{Ω(d)}$, revealing a fundamental barrier even in the distribution-specific setting.
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Submitted 10 March, 2026; v1 submitted 7 October, 2025;
originally announced October 2025.