Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. This translates to a characterization of P.

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   |wiki=    Wicri/Rhénanie
   |area=    UnivTrevesV1
   |flux=    Main
   |étape=   Exploration
   |type=    RBID
   |clé=     ISTEX:93811378C4C92F8FAF325B7FF522DD646CFD34F4
   |texte=   NP-hard sets have many hard instances
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