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On the theory of average case complexity

Authors:

O. GoldreichAuthors Info & Claims

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

Pages 204 - 216

https://doi.org/10.1145/73007.73027

Published: 01 February 1989 Publication History

Abstract

This paper takes the next step in developing the theory of average case complexity initiated by Leonid A. Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. Our results include:

the equivalence of search and decision problems in the context of average case complexity;

an initial analysis of the structure of distributional-NP under reductions which preserve average polynomial-time;

a proof that if all distributional-NP is in average polynomial-time then non-deterministic exponential-time equals deterministic exponential time (i.e., a collapse in the worst case hierarchy);

definitions and basic theorems regarding other complexity classes such as average log-space.

References

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Carter, J., and M. Wegman, "Universal Classes of Hash Functions", JC$$, 1979, Vol. 18, pp. 143-154.

[2]

Cook, S.A., "The Complexity of Theorem Proving Procedures", Proc. 3rd A CM $ymp. on Theory of Computing, pp. 151-158, 1971.

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Garey, M.R., ~nd D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979.

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Goldreich, O., "Towards a Theory of Average Case Complexity (a survey)~', TR-531, Computer Science Department, Technion, Haifa, Israel, March 1988.

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Gurevich, Y. "Complete and Incomplete Randomized NP Problems", Proc. of the 28th IEEE $ym?. on Foundation of Computer Science, 1987, pp. 111-117.

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Gurevich, Y., and D. McC~uley, "Average Case Complete Problems", preprint, 1987.

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Impagliazzo, R., L.A. Levin and M. Luby, "Pseudorandom Number Generation from any One-Way Function", these proceedings, 1989.

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Johnson, D.S., "The NP-Complete Cohmn--an ongoing guide", Jour. of Algorithms, 1984, Vol. 4, pp. 284-299.

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Karp, R.M., "Probabilistic Analysis of Algorithms" manuscript 1986

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Levin, L.A., "Universal Search Problems'', Problemy Peredaci Informacii 9, pp. 115-116, 1973. Translated in problems of Information Transmission 9, pp. 265-266.

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Levin, L.A., "Average Case Complete Problems", SIAM Jour. of Computing, 1986, Vol. 15, pp. 285-286. Extended abstract appeared in Proc. 16th A CM Syrup. on Theory of Computing, 1984, p. 465.

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Levin, L.A., "One-Way Function and Pseudorandom Generators", Proc. 17th A CM Syrup. on Theory of Computing, 1985~ pp. 363-365.

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Levin, L.A., "Homogenous Measures ~ad Polynomial Time Iavaxiaz~ts", Proc. 29th IEEE Syrup. on Foundations of Computer Science, 1988, pp. 36-41.

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Ruzzo, W.L., "Oa Uniform Circuit Complexity", JCS$, 22, 1981, pp. 365- 385.

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Sipser, M., "A Complexity Theoretic Approach to Randomness", Proc. 15th A CM $ymp. on Theory of Computing, 1983, pp. 330-335.

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Venkatesan, R., and L.A. Levin, "Random Instances of a Graph Coloring Problem are Hard", Proc. 20th A CM Syrup. on Theory of Computing, 1988, pp. 217-222.

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Cited By

LEVIN LVENKATESAN R(2018)An Average Case NP-complete Graph Colouring ProblemCombinatorics, Probability and Computing10.1017/S096354831800012327:05(808-828)Online publication date: 2-Apr-2018
https://doi.org/10.1017/S0963548318000123
Vereshchagin N(2013)Improving on Gutfreund, Shaltiel,and Ta-Shma’s Paper “If NP Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances”Computer Science – Theory and Applications10.1007/978-3-642-38536-0_18(203-211)Online publication date: 2013
https://doi.org/10.1007/978-3-642-38536-0_18
Gottesman DIrani S(2009)The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian ProblemsProceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.22(95-104)Online publication date: 25-Oct-2009
https://dl.acm.org/doi/10.1109/FOCS.2009.22
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Index Terms

Recommendations

Average case computational complexity theory
Average-case complexity theory and polynomial-time reductions
Relations between Average-Case and Worst-Case Complexity

The consequences of the worst-case assumption NP=P are very well understood. On the other hand, we only know a few consequences of the analogous average-case assumption “NP is easy on average.” In this paper we establish several new results on the worst-...

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cover image ACM Conferences

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

February 1989

600 pages

ISBN:0897913078

DOI:10.1145/73007

Editor:
D. S. Johnson

Copyright © 1989 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Association for Computing Machinery

New York, NY, United States

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Published: 01 February 1989

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STOC89: 21st Annual ACM Symposium on the Theory of Computing

May 14 - 17, 1989

Washington, Seattle, USA

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STOC '89 Paper Acceptance Rate 56 of 196 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

LEVIN LVENKATESAN R(2018)An Average Case NP-complete Graph Colouring ProblemCombinatorics, Probability and Computing10.1017/S096354831800012327:05(808-828)Online publication date: 2-Apr-2018
https://doi.org/10.1017/S0963548318000123
Vereshchagin N(2013)Improving on Gutfreund, Shaltiel,and Ta-Shma’s Paper “If NP Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances”Computer Science – Theory and Applications10.1007/978-3-642-38536-0_18(203-211)Online publication date: 2013
https://doi.org/10.1007/978-3-642-38536-0_18
Gottesman DIrani S(2009)The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian ProblemsProceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.22(95-104)Online publication date: 25-Oct-2009
https://dl.acm.org/doi/10.1109/FOCS.2009.22
Gutfreund DTa-Shma A(2007)Worst-Case to Average-Case Reductions RevisitedProceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques10.1007/978-3-540-74208-1_41(569-583)Online publication date: 20-Aug-2007
https://dl.acm.org/doi/10.1007/978-3-540-74208-1_41
Gutfreund D(2006)Worst-case vs. algorithmic average-case complexity in the polynomial-time hierarchyProceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation10.1007/11830924_36(386-397)Online publication date: 28-Aug-2006
https://dl.acm.org/doi/10.1007/11830924_36
Durand B(2005)A random NP-complete problem for inversion of 2D cellular automataSTACS 9510.1007/3-540-59042-0_65(95-106)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-59042-0_65
Schuler RYamakami T(2005)Structural average case complexityFoundations of Software Technology and Theoretical Computer Science10.1007/3-540-56287-7_100(128-139)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-56287-7_100
Wang JBelanger J(2005)Honest iteration schemes of randomizing algorithmsAlgorithms and Computation10.1007/3-540-56279-6_79(259-268)Online publication date: 9-Jun-2005
https://doi.org/10.1007/3-540-56279-6_79
Blass AGurevich Y(2005)Randomizing reductions of search problemsFoundations of Software Technology and Theoretical Computer Science10.1007/3-540-54967-6_58(10-24)Online publication date: 31-May-2005
https://doi.org/10.1007/3-540-54967-6_58
Blass AGurevich Y(2005)On the reduction theory for average case complexityComputer Science Logic10.1007/3-540-54487-9_50(17-30)Online publication date: 3-Jun-2005
https://doi.org/10.1007/3-540-54487-9_50
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