A Note on the Sparse Complete Sets

Abstract

Hartmanis and Berman have conjectured that all NP-complete sets are polynomial time isomorphic. A consequence of the conjecture is that there are no sparse NP-complete sets. We show that the existence of an NP-complete set whose complement is sparse implies P = NP. We also show that if there is a polynomial time reduction with sparse range to a PTAPE-complete set, then P=PTAPE. Keywords: reduction, polynomial time, nondeterministic polynomial time, complete sets, sparsity.