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Relations between Average-Case and Worst-Case Complexity

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Abstract

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-case complexity of Arthur-Merlin games (the class AM) under the average-case complexity assumption “NP is easy on average.”

  1. –

    We first consider a stronger notion of “NP is easy on average” namely NP is easy on average with respect to distributions that are computable by polynomial size circuit families. Under this assumption we show that AM can be derandomized to nondeterministic subexponential time.

  2. –

    Under the assumption that NP is easy on average with respect to polynomial-time computable distributions, we show (a) AME=E where AME is the exponential version of AM. This improves an earlier known result that if NP is easy on average then NE=E. (b) For every c>0, \(\mathrm{AM}\subseteq [\mbox{\small io-pseudo}_{\mathrm{NTIME}(n^{c})}]\mathrm{-}\mathrm{NP}\) . Roughly this means that for any language L in AM there is a language L′ in NP so that it is computationally infeasible to distinguish L from L′.

We use recent results from the area of derandomization for establishing our results.

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References

  1. Arvind, V., Köbler, J.: New lowness results for ZPPNP and other complexity classes. J. Comput. Syst. Sci. 65(2), 257–277 (2002)

    Article  MATH  Google Scholar 

  2. Babai, L.: Trading group theory for randomness. In: Proc. 17th Annual ACM Symp. on Theory of Computing, pp. 421–429 (1985)

  3. Babai, L., Moran, S.: Arthur-merlin games: a randomized proof system, and a hierarchy of complexity class. J. Comput. System Sci. 36(2), 254–276 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless exptime has publishable proofs. In: Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pp. 213–219 (1991)

  5. Balcázar, J., Diaz, J., Gabarró, J.: Structural Complexity I. Springer, Berlin (1988)

    MATH  Google Scholar 

  6. Ben-David, S., Chor, B., Goldreich, O., Luby, M.: On the theory of average case complexity. J. Comput. Syst. Sci. 44(2), 193–219 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Book, R.: Tally languages and complexity classes. Inf. Control 26, 186–193 (1974)

    Article  MathSciNet  Google Scholar 

  8. Buhrman, H.: Resource bounded reductions. PhD thesis, University of Amsterdam (1993)

  9. Buhrman, H., Fortnow, L., Pavan, A.: Some results on derandomization. In: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science. LNCS, vol. 2607, pp. 212–222 (2003)

  10. Feige, U., Lund, C.: On the hardness of computing permanent of random matrices. In: Proceedings of 24th Annual ACM Symposium on Theory of Computing, pp. 643–654 (1992)

  11. Geske, J., Huynh, D., Seiferas, J.: A note on almost-everywhere-complex sets and separating deterministic-time-complexity classes. Inf. Comput. 92(1), 97–104 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gemmel, P., Sudan, M.: Higly resilient correctors for polynomials. Inf. Process. Lett. 43, 169–174 (1992)

    Article  Google Scholar 

  13. Gutfreund, D., Shaltiel, R., Ta-Shma, A.: Uniform hardness vs. randomness tradeoffs for Arth ur-Merlin games. Comput. Complex. 12, 85–130 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gurevich, Y.: Average case completeness. J. Comput. Syst. Sci. 42, 346–398 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: exponential time vs. probabilistic polynomial time. J. Comput. Syst. Sci. 65, 672–694 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Impagliazzo, R.: A personal view of average-case complexity theory. In: Proceedings of the 10th Annual Conference on Structure in Complexity Theory, pp. 134–147. IEEE Computer Society Press (1995)

  17. Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 220–229 (1997)

  18. Kabanets, V.: Easiness assumptions and hardness tests: Trading time for zero error. J. Comput. Syst. Sci. 63(2), 236–252 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31, 1501–1526 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Köbler, J., Schuler, R.: Average-case intractability vs. worst-case intractability. Inf. Comput. 190(1), 1–17 (2004)

    Article  MATH  Google Scholar 

  21. Levin, L.: Average case complete problems. SIAM J. Comput. 15, 285–286 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lipton, R.: New directions in testing. In: Distributed Computing and Cryptography. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 2, pp. 191–202. American Mathematics Society, Providence (1991)

    Google Scholar 

  23. Nisan, N., Wigderson, A.: Hardness vs randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  25. Seiferas, J., Fischer, M., Meyer, A.: Separating nondeterministic time complexity classes. J. ACM 25(1), 146–147 (1978)

    MATH  MathSciNet  Google Scholar 

  26. Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudo-random generator. In: 42nd IEEE Symposium on Foundations of Computer Science, pp. 648–657 (2001)

  27. Shaltiel, R., Umans, C.: Pseudorandomness for approximate counting and sampling. In: 20th IEEE Conference on Computational Complexity, pp. 212–226 (2005)

  28. Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. In: Proc. 17th ACM Symp. Theory of Computing, pp. 458–463 (1985)

  29. Žák, S.: A Turing machine time hierarchy. Theor. Comput. Sci. 26, 327–333 (1983)

    Article  Google Scholar 

  30. Wang, J.: Average-case computational complexity theory. In: Hemaspaandra, L., Selman, A. (eds.) Complexity Theory Retrospective II, pp. 295–328. Springer, Berlin (1997)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    A. Pavan

  2. Department of Computer Science and Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    N. V. Vinodchandran

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  1. A. Pavan
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  2. N. V. Vinodchandran
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Correspondence to A. Pavan.

Additional information

A. Pavan research supported by NSF grants CCR-0344817 and CCF-0430807.

N.V. Vinodchandran research supported by NSF grant CCF-0430991, University of Nebraska Layman Award, and Big 12 Fellowship.

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Pavan, A., Vinodchandran, N.V. Relations between Average-Case and Worst-Case Complexity. Theory Comput Syst 42, 596–607 (2008). https://doi.org/10.1007/s00224-007-9071-0

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