Relative-error testing of conjunctions and decision lists

TL;DR

This work advances the understanding of relative-error property testing by showing that natural, sparse Boolean function classes—conjunctions and decision lists—are efficiently testable with relative error. The authors design a one-sided, $O(1/\varepsilon)$-query tester for conjunctions and a near-optimal $\tilde{O}(1/\varepsilon)$-query tester for decision lists, matching the query complexity of best-known standard-model testers up to logarithmic factors. The approach for conjunctions hinges on reducing to anti-monotone conjunction testing and exploiting linear-subspace structure, while the decision-list tester leverages a head-conjunction/tail-decomposition and a gamma-simulation framework to handle relative-error guarantees and imperfect sampling. The results demonstrate that relative-error testing can replicate standard-model efficiencies for these natural classes and provide broader insight into when relative-error testing aligns with or diverges from standard property testing.

Abstract

We study the relative-error property testing model for Boolean functions that was recently introduced in the work of Chen et al. (SODA 2025). In relative-error testing, the testing algorithm gets uniform random satisfying assignments as well as black-box queries to $f$, and it must accept $f$ with high probability whenever $f$ has the property that is being tested and reject any $f$ that is relative-error far from having the property. Here the relative-error distance from $f$ to a function $g$ is measured with respect to $|f^{-1}(1)|$ rather than with respect to the entire domain size $2^n$ as in the Hamming distance measure that is used in the standard model; thus, unlike the standard model, relative-error testing allows us to study the testability of sparse Boolean functions that have few satisfying assignments. It was shown in Chen et al. (SODA 2025) that relative-error testing is at least as difficult as standard-model property testing, but for many natural and important Boolean function classes the precise relationship between the two notions is unknown. In this paper we consider the well-studied and fundamental properties of being a conjunction and being a decision list. In the relative-error setting, we give an efficient one-sided error tester for conjunctions with running time and query complexity $O(1/ε)$. Secondly, we give a two-sided relative-error $\tilde{O}$$(1/ε)$ tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model Bshouty (RANDOM 2020) and Diakonikolas et al. (FOCS 2007).

Theorems & Definitions (79)

• Theorem 1: Relative-error testing of conjunctions
• Theorem 2
• Theorem 3: Relative-error testing of decision lists
• Remark 4: Relative-error testing versus distribution-free testing
• Lemma 5: Approximate triangle inequality for relative distance
• proof
• Theorem 6: Bshouty20
• Theorem 7: DLM+:07
• Lemma 8
• Lemma 9