Randomizing Reductions of Search Problems

Abstract

This paper closes a gap in the foundations of the theory of average-case complexity. First, it clarifies the notion of a feasible solution for a search problem and proves its robustness. Second, it gives a general and usable notion of many–one randomizing reductions of search problems and proves that it has desirable properties. All reductions of search problems to search problems in the literature on average-case complexity can be viewed as such many–one randomizing reductions, including those reductions in the literature that use iterations and therefore do not look many–one. As an illustration, this paper presents a careful proof of a theorem of Impagliazzo and Levin in the framework of the present work.