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Relative-Error Testing of Conjunctions and Decision Lists

Authors Xi Chen , William Pires , Toniann Pitassi , Rocco A. Servedio

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  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property Testing
  • Relative Error

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Abstract

We study the relative-error property testing model for Boolean functions that was recently introduced in the work of [X. Chen et al., 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 2ⁿ 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 [X. Chen et al., 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 Õ(1/ε) tester for decision lists, matching the query complexity of the state-of-the-art algorithm in the standard model [Nader H. Bshouty, 2020; I. Diakonikolas et al., 2007].

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Xi Chen, William Pires, Toniann Pitassi, and Rocco A. Servedio. Relative-Error Testing of Conjunctions and Decision Lists. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.52

Author Details

Xi Chen
  • Columbia University, New York, NY, USA
William Pires
  • Columbia University, New York, NY, USA
Toniann Pitassi
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA

Funding

  • Chen, Xi: Supported by NSF awards CCF-2106429 and CCF-2107187.
  • Pitassi, Toniann: Supported by NSF award CCF-2212136.
  • Servedio, Rocco A.: Supported by NSF awards CCF-2106429 and CCF-2211238.

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