On the relative complexity of hard problems for complexity classes without complete problems

Abstract

We show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound. As a consequence, any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems. A further corollary is that, assuming P ≠ NP, the partial ordering of the polynomial time degrees of NP-sets is not complete, and that there are no degree invariant approximations to NP-complete problems.