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Average case complexity, revisited. (English) Zbl 1343.68114
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 422-450 (2011).
Summary: More than two decades elapsed since Levin set forth a theory of average-case complexity. In this survey we present the basic aspects of this theory as well as some of the main results regarding it. The current presentation deviates from our old [“Notes on Levin’s theory of average-case complexity”, ibid. 6650, 233–247 (2011; Zbl 1343.68111)] in several aspects. In particular:
{\(\bullet\)}
We currently view average-case complexity as referring to the performance on “average” (or rather typical) instances, and not as the average performance on random instances. (Thus, it may be more justified to refer to this theory by the name typical-case complexity, but we retain the name average-case for historical reasons.)
{\(\bullet\)}
We include a treatment of search problems, and a presentation of the reduction of “NP with sampleable distributions” to “NP with P-computable distributions” (due to R. Impagliazzo and L. A. Levin [“No better ways to generate hard NP instances than picking uniformly at random”, in: Proceedings of the 31st annual symposium on foundations of computer science, FOCS 1990. St. Louis, MO: IEEE. 812–821 (1990; doi:10.1109/FSCS.1990.89604)].
{\(\bullet\)}
We include N. Livne’s result [Comput. Complexity 19, No. 4, 477–499 (2010; Zbl 1213.68315)] by which all natural NPC-problems have average-case complete versions. This result seems to shed doubt on the association of P-computable distributions with natural distributions.

For the entire collection see [Zbl 1220.68005].

MSC:
68Q25 Analysis of algorithms and problem complexity
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68-02 Research exposition (monographs, survey articles) pertaining to computer science
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