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Errorless Versus Error-Prone Average-Case Complexity

Authors Shuichi Hirahara, Rahul Santhanam

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Author Details

Shuichi Hirahara
  • Principle of Informatics Research Division, National Institute of Informatics, Tokyo, Japan
Rahul Santhanam
  • Department of Computer Science, University of Oxford, UK

Funding

  • Hirahara, Shuichi: Supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Santhanam, Rahul: Supported by ESPRC New Horizons Grant No. EP/V048201/1 on "Structure vs Randomness in Algorithms and Computation", UK.

Acknowledgements

We thank anonymous reviewers for many insightful and inspiring comments.

Cite As Get BibTex

Shuichi Hirahara and Rahul Santhanam. Errorless Versus Error-Prone Average-Case Complexity. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 84:1-84:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.84

Abstract

We consider the question of whether errorless and error-prone notions of average-case hardness are equivalent, and make several contributions.
First, we study this question in the context of hardness for NP, and connect it to the long-standing open question of whether there are instance checkers for NP. We show that there is an efficient non-uniform non-adaptive reduction from errorless to error-prone heuristics for NP if and only if there is an efficient non-uniform average-case non-adaptive instance-checker for NP. We also suggest an approach to proving equivalence of the two notions of average-case hardness for PH.
Second, we show unconditionally that error-prone average-case hardness is equivalent to errorless average-case hardness for P against NC¹ and for UP ∩ coUP against P.
Third, we apply our results about errorless and error-prone average-case hardness to get new equivalences between hitting set generators and pseudo-random generators.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Interactive proof systems
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • average-case complexity
  • instance checker
  • pseudorandomness

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