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Summary: Reductions Do Not Preserve Fast Convergence
Rates in Average Time
Jay Belanger
A. Pavan
Jie Wang
Abstract
Cai and Selman [CS96] proposed a general definition of average
computation time that, when applied to polynomials, results in a mod-
ification of Levin's [Lev86] notion of average-polynomial-time. The
effect of the modification is to control the rate of convergence of the
expressions that define average computation time. With this modifi-
cation, they proved a hierarchy theorem for average-time complexity
that is as tight as the Hartmanis-Stearns [HS65] hierarchy theorem for
worst-case deterministic time. They also proved that under a fairly
reasonable condition on distributions, called condition W, a distribu-
tional problem is solvable in average-polynomial-time under the mod-
ification exactly when it is solvable in average-polynomial-time under
Levin's definition.
Various notions of reductions, as defined by Levin [Lev86] and
others, play a central role in the study of average-case complexity.
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