 
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 averagepolynomialtime. 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 averagetime complexity
that is as tight as the HartmanisStearns [HS65] hierarchy theorem for
worstcase deterministic time. They also proved that under a fairly
reasonable condition on distributions, called condition W, a distribu
tional problem is solvable in averagepolynomialtime under the mod
ification exactly when it is solvable in averagepolynomialtime under
Levin's definition.
Various notions of reductions, as defined by Levin [Lev86] and
others, play a central role in the study of averagecase complexity.
