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The Hardness of Decision Tree Complexity

Authors Bruno Loff , Alexey Milovanov

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Author Details

Bruno Loff
  • LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal
Alexey Milovanov
  • LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal

Funding

This work was funded by the European Union (ERC, HOFGA, 101041696). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. It was also supported by FCT through the LASIGE Research Unit, ref. UIDB/00408/2025 and ref. UIDP/00408/2025, and by CMAFcIO, FCT Project UIDB/04561/2020, https://doi.org/10.54499/UIDB/04561/2020.

Acknowledgements

The authors would like to thank Wei Zhan for his answer at StackOverflow.

Cite As Get BibTex

Bruno Loff and Alexey Milovanov. The Hardness of Decision Tree Complexity. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 66:1-66:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.66

Abstract

Let f be a Boolean function given as either a truth table or a circuit. How difficult is it to find the decision tree complexity, also known as deterministic query complexity, of f in both cases? We prove that this problem is NC¹-hard and PSPACE-hard, respectively. The second bound is tight, and the first bound is close to being tight.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Decision tree
  • Log-depth circuits

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