__
TR08-022 | 9th January 2008 00:00
__

#### NP-Hard Sets are Exponentially Dense Unless NP is contained in coNP/poly

**Abstract:**
We show that hard sets S for NP must have exponential density, i.e. |S<sub>=n</sub>| ≥ 2<sup>n<sup>ε</sup></sup> 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<sup>1-ε</sup> 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<sup>δ</sup> for some δ > 0. This result also holds for Turing reductions that make n<sup>1-ε</sup> queries.