Abstract:
We show that hard sets S for NP must have exponential density, i.e. |S_=n| ≥ 2^nε for some ε > 0 and infinitely many n, unless coNP ⊆ NP\poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n^1-ε queries. In addition we study the instance complexity of NP-hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n^δ for some δ > 0. This result also holds for Turing reductions that make n^1-ε queries.