A Note on 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 $P = PTAPE$.