Complexity Classes of Optimization Functions

Abstract

In this paper, complexity classes of functions defined via taking maxima or minima (cf. the work of Krentel) or taking middle elements (cf. the work of Toda) are examined. A number of axioms for a class to be a so-called p-founded class of optimization functions are given. it is shown that many natural function classes fulfill these axioms, Then these classes are examined concerning their relationship to complexity classes of sets. To this end, complexity preserving operators S and F for encoding function classes into set classes and vice versa are introduced. It is shown how these operators translate closure properties from one class to another, how they relate operators on classes of functions and classes of sets, and how they encode classes of maximum, minimum, or median functions into well-studied classes of sets. The fixpoints of the compositional operator F · S are examined and shown to be exactly those function classes "closed under binary search." Let F1 and F2 be two such fixpoints, and K1 and K2 be their corresponding classes of sets (i.e., their images under the operator S). Then F1 ⊆ F2 if and only if K1 ⊆ K2, and F1 = F2 if and only if K1 = K2. A number of natural classes of functions are shown to be such fixpoints. Thus we build hierachies of function classes with the same inclusional relationships as the polynomial time hierarchy of sets and the counting hierarchy of sets, and we prove many interesting structural properties of these hierarchies.