NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly
Harry Buhrman, John M. Hitchcock · 2008
We show that hard sets S for NP must have exponential density, i.e. |S=n| ges 2nepsifor some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1-isinqueries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n1-isinqueries.