The density and complexity of polynomial cores for intractable sets

Abstract

Let A be a recursive problem not in P. Lynch has shown that A then has an infinite recursive polynomial complexity core. This is a collection C of instances of A such that every algorithm deciding A needs more than polynomial time almost everywhere on C. We investigate the complexity of recognizing the instances in such a core, and show that every recursive problem A not in P has an infinite core recognizable in subexponential time. We further study how dense the core sets for A can be, under various assumptions about the structure of A. Our main results in this direction are that if P ≠ NP, then NP-complete problems have polynomially nonsparse cores recognizable in subexponential time, and that EXPTIME-complete problems have cores of exponential density recognizable in exponential time.