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Paper:

TR08-022 | 9th January 2008 00:00

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

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TR08-022
Authors: Harry Buhrman, John Hitchcock
Publication: 11th March 2008 07:16
Downloads: 1649
Keywords: 


Abstract:

We show that hard sets S for NP must have exponential density, i.e. |S<sub>=n</sub>| &#8805; 2<sup>n<sup>&#949;</sup></sup> for some &#949; > 0 and infinitely many n, unless coNP &#8838; NP\poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n<sup>1-&#949;</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>&#948;</sup> for some &#948; > 0. This result also holds for Turing reductions that make n<sup>1-&#949;</sup> queries.



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