# Average-case Complexity

@article{Bogdanov2008AveragecaseC, title={Average-case Complexity}, author={Andrej Bogdanov and Luca Trevisan}, journal={2008 49th Annual IEEE Symposium on Foundations of Computer Science}, year={2008}, pages={11-11} }

We review the many open questions and the few things that are known about the average-case complexity of computational problems. We shall follow the presentations of Impagliazzo, of Goldreich, and of Bogdanov and the author, and focus on the following subjects. (i). Average-case tractability. What does it mean for a problem to have an "efficient on average'' algorithm with respect to a distribution of instances? There is more than one ``correct'' answer to this question, and a numberof…

## 117 Citations

Special Issue On Worst-case Versus Average-case Complexity Editors’ Foreword

- Computer Sciencecomputational complexity
- 2007

This special issue aims to present a small sample of papers that are representative of the different types of results that have been obtained in average-case complexity, e.g., whether NP = P implies that NP has problems that are hard on average.

Average Case Complexity, Revisited

- MathematicsStudies in Complexity and Cryptography
- 2011

This survey presents the basic aspects of this theory as well as some of the main results regarding it, and includes Livne's result by which all natural NPC-problems have average-case complete versions, which seems to shed doubt on the association of P-computable distributions with natural distributions.

Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2022

A probabilistic theory of meta-complexity is developed, by incorporating randomness into the notion of complexity of a string x, through a new probabilism variant of time-bounded Kolmogorov complexity that is called pK complexity.

Average-case hardness of NP from exponential worst-case hardness assumptions

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2021

This paper analyzes average-case complexity through the lens of worst-case meta-complexity using a new “algorithmic” proof of language compression and weak symmetry of information for time-bounded Kolmogorov complexity and presents a new notion of universal heuristic scheme.

Non-Black-Box Worst-Case to Average-Case Reductions within NP

- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

This paper presents the first non-black-box worst-case to average-case reduction from a problem outside coNP (unless Random 3SAT is easy for coNP algorithms) to a distributional NP problem and proposes a research program for excluding Heuristica, i.e., establishing an equivalence between the worst- case and average- case hardness of NP through the lens of MINKT or the Minimum Circuit Size Problem (MCSP.

Finding Errorless Pessiland in Error-Prone Heuristica

- Computer ScienceCCC
- 2022

Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence…

Errorless Versus Error-Prone Average-Case Complexity

- Computer Science, MathematicsITCS
- 2022

It is shown unconditionally that error-prone average-case hardness is equivalent to errorless average- case hardness for P against NC1 and for UP ∩ coUP against P, and applied to get new equivalences between hitting set generators and pseudo-random generators.

Average-case Hardness of NP and PH from Worst-case Fine-grained Assumptions

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2021

This paper shows that several standard worst-case complexity assumptions are sufﬁcient to imply non-trivial average-case hardness of NP or PH, and constructs quite efﷁcient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest.

Average-Case Completeness in Tag Systems

- Computer ScienceSTACS
- 2019

This work shows that a tag system can efficiently simulate a Turing machine even when the input is provided in an extremely simple encoding which adds just log n carefully set bits to encode an arbitrary Turing machine input of length n.

One-Way Functions and a Conditional Variant of MKTP

- Mathematics, Computer ScienceFSTTCS
- 2021

It is shown that McKTP is hard-on-average if and only if there are logspace-computable OWFs, which means the existence of OWFs is inextricably linked to the average-case hardness of this NP -complete problem.

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