On the Structure of Sets in NP and Other Complexity Classes

The relationship between resource bounded deterministic and nondeterministic complexity classes has been extensively studied. For polynomial time, the associated question P = NP? is particularly important because of the large number of problems of practical interest that can be solved by nondeterministic polynomial time bounded devices. If P = NP, then these psoblems all have determinisitic polynomial time solutiolns whereas otherwise only exponential time solutions exist. Furthermore, the class NP contains complete problen:s to which all other members of NP can be reduced [S, ?,16] and P = NIP if and only if one of these complete problems is in P. With few exceptions, most intuitively appealk?g members of NP have been shown to be complete. In this paper, we study the structure of sets in NP. We develop a number of simple tools which facilitate the study o.? the relatke complexity 04 sets with res,pect to polynomial time reducibility. The method yields the existence of a minimal pair (with respect to P) of sets A,, S E NP which ;\re not complete. This strengthens earlier results “of Ladner [lo] (minimill pair-ncb upper baound) and Machtey Cl!51 (minimal pair-subexponential but not .qecessarily in NP). In addition, the method can be used to constru@ partial orders of degrees with respect to polynomial time reducibility.