Reductions Do Not Preserve Fast Convergence Rates in Average Time

Abstract

Cai and Selman [CS] proposed the following definition for measuring average computation time: A time function t is T on average over a distribution μ if, for all $ n \geq 1 $ , $ \sum_{|x|\geq n}T^{-1}(t(x))|x|^{-1}\mu (x) \leq W_n $ , where $ W_n = \mu (\{ x : |x| \geq n\} ) $ . This definition results in a modification of Levin's notion of average time [L]. The effect of the modification is to control the rate of convergence of the expressions that define average computation time. With this modification, they proved a hierarchy theorem for average-time complexity that is as tight as the Hartmanis—Stearns [HS] hierarchy theorem for worst-case deterministic time. They also proved that under a fairly reasonable condition on distributions, called condition W, a distributional problem is solvable in average polynomial time under the modification exactly when it is solvable in average polynomial time under Levin's definition. Various notions of reductions, as defined by Levin [L] and others, play a central role in the study of average-case complexity. However, the class of distributional problems that are solvable in average polynomial time under the modification is not closed under the standard reductions. In particular, we prove that there is a distributional problem that is not solvable in average polynomial time under the modification but is reducible, by the identity function, to a distributional problem that is, and whose distribution even satisfies condition W.