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Relative-Error Unateness Testing

Authors Xi Chen , Diptaksho Palit , Kabir Peshawaria , William Pires , Rocco A. Servedio , Yiding Zhang

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

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Abstract

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [X. Chen et al., 2025; Chen et al., 2025; Chen et al., 2025]. In this paper we consider the problem of relative-error testing an unknown and arbitrary f: {0,1}ⁿ → {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 Õ(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 Õ(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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Xi Chen, Diptaksho Palit, Kabir Peshawaria, William Pires, Rocco A. Servedio, and Yiding Zhang. Relative-Error Unateness Testing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.62

Author Details

Xi Chen
  • Columbia University, New York, NY, USA
Diptaksho Palit
  • Boston University, MA, USA
Kabir Peshawaria
  • Boston University, MA, USA
William Pires
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA
Yiding Zhang
  • Boston University, MA, USA

Funding

  • Chen, Xi: X.C. is funded by NSF CCF-2106429, CCF-2107187, and OpenAI through the UK AI Security Institute’s Alignment Project.
  • Palit, Diptaksho: D.P. is funded by NSF DMS-2022446.
  • Pires, William: W.P. is funded by NSF CCF-2106429 and CCF-2107187.
  • Servedio, Rocco A.: R.A.S. is funded by NSF CCF-2211238, NSF CCF-2106429, and OpenAI through the UK AI Security Institute’s Alignment Project.
  • Zhang, Yiding: Y.Z. is funded by NSF GCR-2428487.

Acknowledgements

D.P., K.P., and Y.Z. started working on this project for a course on Sublinear Algorithms taught by Sofya Raskhodnikova [Sofya Raskhodnikova, 2025]. Part of this work was carried out while visiting the Simons Institute for the Theory of Computing at UC Berkeley.

References

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