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On the Pseudo-Deterministic Query Complexity of NP Search Problems

Authors Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, Rahul Santhanam

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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Proof complexity
Keywords
  • Pseudo-determinism
  • Query complexity
  • Proof complexity

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Abstract

We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.

Cite As Get BibTex

Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the Pseudo-Deterministic Query Complexity of NP Search Problems. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.36

Author Details

Shafi Goldwasser
  • University of California, Berkeley, CA, USA
Russell Impagliazzo
  • University of California, San Diego, CA, USA
Toniann Pitassi
  • University of Toronto, Canada
  • Columbia University, New York, NY, USA
  • Institute of Advanced Study, Princeton, NJ, USA
Rahul Santhanam
  • University of Oxford, UK

Funding

  • Goldwasser, Shafi: Research supported in part by DARPA under Contract No. HR001120C0015.
  • Impagliazzo, Russell: Research supported by NSF and the Simons Foundation.
  • Pitassi, Toniann: Research supported by NSERC, the IAS School of Mathematics and NSF Grant No. CCF-1900460.

Acknowledgements

We thank Ofer Grossman, Ran Raz, Avi Wigderson and Ryan Williams for helpful discussions. The quantum query upper bound for FIND1 was pointed out to the fourth author by Igor Oliveira. We also thank the anonymous CCC reviewers for very helpful comments.

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