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Reductions and convergence rates of average time

  • Jay Belanger
  • Jie Wang
Session 9
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1090)

Abstract

Using a fast convergence rate to measure computation time on average has recently been investigated [CS95a], which modifies the notion of T-time on average [BCGL92]. This modification admits an average-case time hierarchy which is independent of distributions and is as tight as the Hartmanis-Sterns hierarchy for the worst-case deterministic time [HS65]. Various notions of reductions, defined by Levin [Lev86] and others, have played a central role in studying average-case complexity. However, unless the class of admissible distributions is restricted, these notions of reductions cannot be applied to the modified definition. In particular, we show that under the modified definition, there exists a problem which is not computable in average polynomial time, but is efficiently reducible to one that is. We hope that this observation can further stimulate research on finding suitable reductions in this new line of investigation.

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References

  1. [BCGL92]
    S. Ben-David, B. Chor, O. Goldreich, and M. Luby. On the theory of average case complexity. J. Comp. Sys. Sci., 44:193–219, 1992. (First appeared in Proc. 21st STOC, ACM, pages 204–216, 1989.)Google Scholar
  2. [BG95]
    A. Blass and Y. Gurevich. Matrix transformation is complete for the average case. SIAM J. Comput., 24:3–29, 1995.Google Scholar
  3. [BW]
    J. Belanger and J. Wang. No NP problems averaging on ranking of distributions are harder. Submitted.Google Scholar
  4. [BW95]
    J. Belanger and J. Wang. Rankable distributions do not provide harder instances than uniform distributions. Proc. 1st COCOON, vol 959 of Lect. Notes in Comp. Sci., pages 410–419, 1995.Google Scholar
  5. [CS95a]
    J.-Y. Cai and A. Selman. Average time complexity classes. Elect. Col. Comp. Complexity TR95-019, 1995.Google Scholar
  6. [CS95b]
    J.-Y. Cai and A. Selman. Personal communication.Google Scholar
  7. [GGH94]
    M. Goldmann, P. Grape, and J. Håstad. On average time hierarchies. Inf. Proc. Lett., 49:15–20, 1994.Google Scholar
  8. [GHS91]
    J. Geske, D. Huynh and J. Seiferas. A note on almost-everywhere complex sets with application to polynomial complexity degrees. Inf. and Comput., 92(1):97–104, 1991.CrossRefGoogle Scholar
  9. [GS87]
    Y. Gurevich and S. Shelah. Expected Computation Time for Hamiltonian Path Problem. SIAM J. on Computing, 16:3(1987), pp. 486–502.Google Scholar
  10. [Gur89]
    Y. Gurevich. The challenger-solver game: variations on the theme of P =? NP. EATCS Bulletin, pages 112–121, 1989.Google Scholar
  11. [Gur91]
    Y. Gurevich. Average case completeness. J. Comp. Sys. Sci., 42:346–398, 1991.Google Scholar
  12. [Har11]
    G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., 10:54–90, 1911.Google Scholar
  13. [HS65]
    J. Hartmanis and R. Stearns. On the computational complexity of algorithms. Trans. Amer. Math. Soc., 117:285–306, 1965.Google Scholar
  14. [Joh84]
    D. Johnson. The NP-completeness column: an ongoing guide. Journal of Algorithms, 5:284–299, 1984.CrossRefGoogle Scholar
  15. [Imp95]
    R. Impagliazzo. A personal view of average-case complexity. Proc. 10th Structures, IEEE, pages 134–147, 1995.Google Scholar
  16. [Lev86]
    L. Levin. Average case complete problems. SIAM J. Comput., 15:285–286, 1986. (First appeared in Proc. 16th STOC, ACM, page 465, 1984.)CrossRefGoogle Scholar
  17. [Ven91]
    R. Venkatesan. Average-Case Intractability. Ph.D. Thesis (Advisor: L. Levin), Boston University, 1991.Google Scholar
  18. [VL88]
    R. Venkatesan and L. Levin. Random instances of a graph coloring problem are hard. In Proc. 20th STOC, pages 217–222, 1988.Google Scholar
  19. [VR92]
    R. Venkatesan and S. Rajagopalan. Average case intractability of diophantine and matrix problems. In Proc. 24th STOC, pages 632–642, 1992.Google Scholar
  20. [Wan]
    J. Wang. Average-case computational complexity theory. Complexity Theory Retrospective II (A. Selman and L. Hemaspaandra eds), Springer-Verlag, to appear. (Also available by anonymous ftp at ftp.uncg.edu under the directory people/wangjie with file name avg_comp.ps.gz.)Google Scholar
  21. [Wan95]
    J. Wang. Average-case completeness of a word problem for groups. In Proc. 27th STOC, pages 325–334, 1995.Google Scholar
  22. [WB93]
    J. Wang and J. Belanger. On average-P vs. average-NP. In K. Ambos-Spies, S. Homer, and U. Schönings, editors, Complexity Theory—Current Research, pages 47–67. Cambridge University Press, 1993.Google Scholar
  23. [WB95]
    J. Wang and J. Belanger. On the NP-isomorphism problem with respect to random instances. J. Comp. Sys. Sci., 50(1995), pp. 151–164.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of North Carolina at GreensboroGreensboroUSA
  2. 2.Division of Mathematics and Computer ScienceNortheast Missouri State UniversityKirksvilleUSA

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