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Bounds for Hardness Condensation in the Query Model

Authors Chandrima Kayal , Rajat Mittal , Sai Soumya Nalli , Manaswi Paraashar , Karthikeya Polisetty, Jayalal Sarma , Nitin Saurabh

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ACM Subject Classification
  • Theory of computation → Oracles and decision trees
Keywords
  • Query Complexity
  • Decision Trees
  • Hardness Condensation

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Abstract

For any Boolean function f:{0,1}ⁿ → {0,1} with a complexity measure having value k ≪ n, is it possible to restrict the function f to Θ(k) variables while keeping the complexity preserved at Θ(k)? Instantiation of this question for the measure of circuit complexity of the Boolean function was shown to be related to circuit lower bounds (Buresh-Oppenheim and Santhanam, 2006). Variants of the above question were also shown to have connections to the log-rank conjecture in communication complexity (Hrubeš, 2024) and lower bounds in proof complexity (Razborov, 2016). In the context of communication and query complexity, this question was recently studied by Göös, Newman, Riazanov and Sokolov (2024). They showed, among other results, that query complexity cannot be condensed losslessly. 
In this work, we show that there exists a Boolean function f such that any restriction of f to O(ℳ(f)) variables has ℳ(⋅)-complexity at most Õ(ℳ(f)^{2/3}), where ℳ is one of block sensitivity (bs), fractional block sensitivity (fbs), certificate complexity (𝖢), deterministic query complexity (𝖣), zero-error randomized query complexity (𝖱₀), and AND (and OR)-decision tree query complexity. This improves upon the results of Göös, Newman, Riazanov, and Sokolov (2024) for 𝖣 and 𝖱₀, and in particular answers their open question about the condensation of block sensitivity.
We complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function f there exists a restriction of f to O(ℳ(f)) variables such that its ℳ(⋅)-complexity is at least Ω(ℳ(f)^{1/2}), where ℳ ∈ {bs,fbs,𝖢,UC_{min},UC₁,UC,𝖣,deg̃,λ}. In addition, we show lossy condensation for randomized and quantum query complexity with a slightly smaller exponent.

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Chandrima Kayal, Rajat Mittal, Sai Soumya Nalli, Manaswi Paraashar, Karthikeya Polisetty, Jayalal Sarma, and Nitin Saurabh. Bounds for Hardness Condensation in the Query Model. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.30

Author Details

Chandrima Kayal
  • Université Paris Cité, CNRS, IRIF, Paris, France
Rajat Mittal
  • Indian Institute of Technology Kanpur, India
Sai Soumya Nalli
  • Microsoft Research India, Bangalore, India
Manaswi Paraashar
  • Indian Institute of Technology Hyderabad, India
Karthikeya Polisetty
  • Indian Institute of Technology Madras, India
Jayalal Sarma
  • Indian Institute of Technology Madras, India
Nitin Saurabh
  • Indian Institute of Technology Hyderabad, India

Funding

  • Kayal, Chandrima: Funded by French PEPR integrated project EPiQ (ANR-22-PETQ-0007) and ANR Grant FLITTLA (ANR-21-CE48-0023).
  • Saurabh, Nitin: Supported by ANRF ARG-MATRICS grant ANRF/ARGM/2025/003191/MTR.

Acknowledgements

CK and NS would like to thank Meena Mahajan and Gaurav Sood for insightful discussions during the initial stages of the work. We would also like to thank anonymous reviewers whose careful reading and thoughtful comments greatly improved the presentation of this paper. We would also like to thank the audience in STCS seminar at TIFR, Mumbai for helpful discussions.

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