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}
}
• Published 8 June 2006
• Mathematics
• 2008 49th Annual IEEE Symposium on Foundations of Computer Science
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 Science
computational 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
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 Science
Electron. 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
• Shuichi Hirahara
• Computer Science, Mathematics
Electron. 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
• Shuichi Hirahara
• Computer Science, Mathematics
2018 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 Science
CCC
• 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, Mathematics
ITCS
• 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 Science
Electron. 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 Science
STACS
• 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 Science
FSTTCS
• 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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