Rankable distributions do not provide harder instances than uniform distributions

  • Jay Belanger
  • Jie Wang
Session 7B: Complexity Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)


We show that polynomially rankable distributions, proposed in [RS93], do not provide harder instances than uniform distributions for NP problems. In particular, we show that if Levin's randomized tiling problem is solvable in polynomial time on average, then every NP problem under any p-rankable distribution is solvable in average polynomial time with respect to rankability. One of the motivations for polynomially rankable distributions was to get average-case hierarchies, and we present a reasonably tight hierarchy result for average-case complexity classes under p-time computable distributions.


Turing Machine Ranking Function Deterministic Algorithm Rankable Distribution Hard Instance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.CrossRefGoogle Scholar
  2. [BG]
    A. Blass and Y. Gurevich. Randomizing reductions of decision problems (tentative title). Personal communication.Google Scholar
  3. [BG93]
    A. Blass and Y. Gurevich. Randomizing reductions of search problems. SIAM J. Comput., 22:949–975, 1993.CrossRefGoogle Scholar
  4. [BG94]
    A. Blass and Y. Gurevich. Matrix decomposition is complete for the average case. SIAM J. Comput., 1994. to appear.Google Scholar
  5. [BH77]
    L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets. SIAM J. Comput., 6:305–321, 1977.Google Scholar
  6. [Bol85]
    B. Bollobás. Random Graphs. Academic Press, 1985.Google Scholar
  7. [CS95]
    J.-Y. Cai and A. Selman. Average time complexity classes. Manuscript.Google Scholar
  8. [GGH94]
    M. Goldmann, P. Grape, and J. Hastad. On average time hierarchies. Inf. Proc. Lett., 49:15–20, 1994.CrossRefGoogle Scholar
  9. [GS87]
    Y. Gurevich and S. Shelah. Expected computation time for hamiltonian path problem. SIAM J. Comput., 16:486–502, 1987.CrossRefGoogle Scholar
  10. [Gur91]
    Y. Gurevich. Average case completeness. J. Comp. Sys. Sci., 42:346–398, 1991.CrossRefGoogle Scholar
  11. [Har11]
    G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., (2),10:54–90, 1911.Google Scholar
  12. [HS65]
    J. Hartmanis and R. Sterns. On the computational complexity of algorithms. Trans. Amer. Math. Soc., 117:285–306, 1965.Google Scholar
  13. [IL90]
    R. Impagliazzo and L. Levin. No better ways to generate hard NP instances than picking uniformly at random. In Proc. 31st FOCS, pages 812–821, 1990.Google Scholar
  14. [Jai91]
    R. Jain. The Art of Computer Systems Performance Analysis. John Wiley & Sons, 1991.Google Scholar
  15. [JK69]
    N. Johnson and S. Kotz. Distributions in Statistics-Discrete Distributions. John Wiley & Sons, 1969.Google Scholar
  16. [Joh84]
    D. Johnson. The NP-completeness column: an ongoing guide. Journal of Algorithms, 5:284–299, 1984.CrossRefGoogle Scholar
  17. [Lev86]
    L. Levin. Average case complete problems. SIAM J. Comput., 15:285–286, 1986.CrossRefGoogle Scholar
  18. [LV92]
    M. Li and P. Vitányi. Average case complexity under the universal distribution equals worst-case complexity. Inf. Proc. Lett., 42:145–149, 1992.CrossRefGoogle Scholar
  19. [RS93]
    R. Reischuk and C. Schindelhauer. Precise average case complexity. STACS'93, vol 665 of Lect. Notes in Comp. Sci., pages 650–661, 1993.Google Scholar
  20. [SY92]
    R. Schuler and T. Yamakami. Structural average case complexity. FSTTCS'92, vol 652 of Lect. Notes in Comp. Sci., pages 128–139, 1992.Google Scholar
  21. [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
  22. [VR92]
    R. Venkatesan and S. Rajagopalan. Average case intractability of diophantine and matrix problems. In Proc. 24th STOC, pages 632–642, 1992.Google Scholar
  23. [Wan95]
    J. Wang. Average-case completeness of a word problem for groups. In Proc. 27th STOC, 1995. To appear.Google Scholar
  24. [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
  25. [WB95]
    J. Wang and J. Belanger. On the NP-isomorphism problem with respect to random instances. J. Comp. Sys. Sci., 50:151–164, 1995.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Jay Belanger
    • 1
  • Jie Wang
    • 2
  1. 1.Div. of Math & Computer ScienceNortheast Missouri State Univ.Kirksville
  2. 2.Dept. of Mathematical SciencesUniv. of North Carolina at GreensboroGreensboro

Personalised recommendations