# Reductions and convergence rates of average time

## 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [BG95]A. Blass and Y. Gurevich. Matrix transformation is complete for the average case.
*SIAM J. Comput.*, 24:3–29, 1995.Google Scholar - [BW]J. Belanger and J. Wang. No NP problems averaging on ranking of distributions are harder. Submitted.Google Scholar
- [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 - [CS95a]J.-Y. Cai and A. Selman. Average time complexity classes.
*Elect. Col. Comp. Complexity*TR95-019, 1995.Google Scholar - [CS95b]J.-Y. Cai and A. Selman. Personal communication.Google Scholar
- [GGH94]M. Goldmann, P. Grape, and J. Håstad. On average time hierarchies.
*Inf. Proc. Lett.*, 49:15–20, 1994.Google Scholar - [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 - [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 - [Gur89]Y. Gurevich. The challenger-solver game: variations on the theme of P =? NP.
*EATCS Bulletin*, pages 112–121, 1989.Google Scholar - [Gur91]Y. Gurevich. Average case completeness.
*J. Comp. Sys. Sci.*, 42:346–398, 1991.Google Scholar - [Har11]G. Hardy. Properties of logarithmico-exponential functions.
*Proc. London Math. Soc.*, 10:54–90, 1911.Google Scholar - [HS65]J. Hartmanis and R. Stearns. On the computational complexity of algorithms.
*Trans. Amer. Math. Soc.*, 117:285–306, 1965.Google Scholar - [Joh84]D. Johnson. The NP-completeness column: an ongoing guide.
*Journal of Algorithms*, 5:284–299, 1984.CrossRefGoogle Scholar - [Imp95]R. Impagliazzo. A personal view of average-case complexity.
*Proc. 10th Structures*, IEEE, pages 134–147, 1995.Google Scholar - [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 - [Ven91]R. Venkatesan.
*Average-Case Intractability*. Ph.D. Thesis (Advisor: L. Levin), Boston University, 1991.Google Scholar - [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 - [VR92]R. Venkatesan and S. Rajagopalan. Average case intractability of diophantine and matrix problems. In
*Proc. 24th STOC*, pages 632–642, 1992.Google Scholar - [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 - [Wan95]J. Wang. Average-case completeness of a word problem for groups. In
*Proc. 27th STOC*, pages 325–334, 1995.Google Scholar - [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 - [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