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.

• Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (College Park, Maryland, June 22-26, 2008), pages 1-7. IEEE Computer Society Press, 2008.
  [DOI link | PDF | PS]

• Technical Report TR08-022, Electronic Colloquium on Computational Complexity, 2008.