Query Complexity of Search Problems

Authors Arkadev Chattopadhyay , Yogesh Dahiya , Meena Mahajan

Thumbnail PDF


  • Filesize: 0.78 MB
  • 15 pages

Document Identifiers

Author Details

Arkadev Chattopadhyay
  • Tata Institute of Fundamental Research, Mumbai, India
Yogesh Dahiya
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India
Meena Mahajan
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India

Cite AsGet BibTex

Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan. Query Complexity of Search Problems. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we provide the following improvements upon the known relationship between pseudo-deterministic and deterministic query complexity for total search problems: - We show that deterministic query complexity is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13). - We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) we exhibit an exp(Ω̃(n^{1/4})) separation for the SearchCNF relation for random k-CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of SearchCNF associated with random k-CNFs. (2) we exhibit an exp(Ω(n)) separation for the ApproxHamWt relation. The previous best known separation for any relation was exp(Ω(n^{1/2})). We also separate pseudo-determinism from randomness in And and (And,Or) decision trees, and determinism from pseudo-determinism in Parity decision trees. For a hypercube colouring problem, that was introduced by Goldwasswer, Impagliazzo, Pitassi and Santhanam (CCC'21) to analyze the pseudo-deterministic complexity of a complete problem in TFNP^{dt}, we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is Ω(n^{1/3}); Goldwasser et al. showed an Ω(n^{1/2}) bound for general block-sensitivity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Models of computation
  • Decision trees
  • Search problems
  • Pseudo-determinism
  • Randomness


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - Resolution made simple. J. ACM, 48(2):149-169, 2001. URL: https://doi.org/10.1145/375827.375835.
  2. Harry Buhrman and Ronald De Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. Google Scholar
  3. Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan. Query complexity of search problems. Electron. Colloquium Comput. Complex., TR23-039, 2023. URL: https://eccc.weizmann.ac.il/report/2023/039/.
  4. Arkadev Chattopadhyay, Yogesh Dahiya, Nikhil S Mande, Jaikumar Radhakrishnan, and Swagato Sanyal. Randomized versus deterministic decision tree size. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 867-880, 2023. Google Scholar
  5. Daniel Dadush and Samarth Tiwari. On the Complexity of Branching Proofs. In Shubhangi Saraf, editor, 35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 34:1-34:35, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.34.
  6. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 902-911, 2018. Google Scholar
  7. Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electron. Colloquium Comput. Complex., page 136, 2011. Google Scholar
  8. Oded Goldreich, Shafi Goldwasser, and Dana Ron. On the possibilities and limitations of pseudodeterministic algorithms. In Robert D. Kleinberg, editor, Innovations in Theoretical Computer Science ITCS, pages 127-138. ACM, 2013. See also ECCC Vol. 19, T.R. 12-101, 2012. Google Scholar
  9. Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the pseudo-deterministic query complexity of NP search problems. In Valentine Kabanets, editor, 36th Computational Complexity Conference CCC, volume 200 of LIPIcs, pages 36:1-36:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  10. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
  11. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Guest column: Models of computation between decision trees and communication. ACM SIGACT News, 52(2):46-70, 2021. Google Scholar
  12. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Log-rank and lifting for and-functions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 197-208, 2021. Google Scholar
  13. László Lovász, Moni Naor, Ilan Newman, and Avi Wigderson. Search problems in the decision tree model. SIAM Journal on Discrete Mathematics, 8(1):119-132, 1995. Google Scholar
  14. Noam Nisan. Crew prams and decision trees. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 327-335, 1989. Google Scholar
  15. Andrew Chi-Chih Yao. Lower bounds by probabilistic arguments (extended abstract). In 24th Annual Symposium on Foundations of Computer Science FOCS, pages 420-428. IEEE Computer Society, 1983. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail