The Structure of Complete Degrees

The notion of NP-completeness has cut across many elds and has provided a means of identifying deep and unexpected commonalities. Problems from areas as diverse as combinatorics, logic, and operations research turn out to be NP-complete and thus computationally equivalent in the sense discussed in the next paragraph. PSPACE-completeness, NEXP-completeness, and completeness for other complexity classes have likewise been used to show commonalities in a variety of other problems. This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural diierences between NP-complete sets, what are they and what explains the diierences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need rst to formalize NP-completeness. There are a number of competing deenitions of NP-completeness. (See Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time many-one reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: