Average case complexity, revisited

Abstract

More than two decades elapsed since Levin set forth a theory of average-case complexity. In this survey we present the basic aspects of this theory as well as some of the main results regarding it. The current presentation deviates from our old "Notes on Levin's Theory of Average-Case Complexity" (ECCC, TR97-058, 1997) in several aspects. In particular: We currently view average-case complexity as referring to the performance on "average" (or rather typical) instances, and not as the average performance on random instances. (Thus, it may be more justified to refer to this theory by the name typical-case complexity, but we retain the name average-case for historical reasons.) We include a treatment of search problems, and a presentation of the reduction of "NP with sampleable distributions" to "NP with P-computable distributions" (due to Impagliazzo and Levin, 31st FOCS, 1990). We include Livne's result (ECCC, TR06-122, 2006) by which all natural NPC-problems have average-case complete versions. This result seems to shed doubt on the association of P-computable distributions with natural distributions. © 2011 Springer-Verlag Berlin Heidelberg.