Relative-error testing of conjunctions and decision lists
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912322193195008 |
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| author | Chen, Xi Pires, William Pitassi, Toniann Servedio, Rocco A. |
| author_facet | Chen, Xi Pires, William Pitassi, Toniann Servedio, Rocco A. |
| contents | 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}$$(1/ε)$ tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model Bshouty (RANDOM 2020) and Diakonikolas et al. (FOCS 2007). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative-error testing of conjunctions and decision lists Chen, Xi Pires, William Pitassi, Toniann Servedio, Rocco A. Computational Complexity Discrete Mathematics Data Structures and Algorithms We study the relative-error property testing model for Boolean functions that was recently introduced in the work of Chen et al. (SODA 2025). In relative-error testing, the testing algorithm gets uniform random satisfying assignments as well as black-box queries to $f$, and it must accept $f$ with high probability whenever $f$ has the property that is being tested and reject any $f$ that is relative-error far from having the property. Here the relative-error distance from $f$ to a function $g$ is measured with respect to $|f^{-1}(1)|$ rather than with respect to the entire domain size $2^n$ as in the Hamming distance measure that is used in the standard model; thus, unlike the standard model, relative-error testing allows us to study the testability of sparse Boolean functions that have few satisfying assignments. It was shown in Chen et al. (SODA 2025) that relative-error testing is at least as difficult as standard-model property testing, but for many natural and important Boolean function classes the precise relationship between the two notions is unknown. In this paper we consider the well-studied and fundamental properties of being a conjunction and being a decision list. In the relative-error setting, we give an efficient one-sided error tester for conjunctions with running time and query complexity $O(1/ε)$. Secondly, we give a two-sided relative-error $\tilde{O}$$(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). |
| title | Relative-error testing of conjunctions and decision lists |
| topic | Computational Complexity Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.08987 |