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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.62

Relative-Error Unateness Testing

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

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

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)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2026.62,
  author =	{Chen, Xi and Palit, Diptaksho and Peshawaria, Kabir and Pires, William and Servedio, Rocco A. and Zhang, Yiding},
  title =	{{Relative-Error Unateness Testing}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{62:1--62:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.62},
  URN =		{urn:nbn:de:0030-drops-264511},
  doi =		{10.4230/LIPIcs.ICALP.2026.62},
  annote =	{Keywords: Property Testing, Relative Error}
}

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Relative-Error Unateness Testing

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