Abstract
It is known that any set A not in P contains an infinite complexity core, that is, a set C \(\subseteq \) A such that any algorithm for A takes superpolynomial time almost everywhere on C. We investigate the conditions under which an intractable set A can possess a core that is maximal with respect to inclusion; such a core could be understood as containing exactly the “inherently hard“ instances of A. We show that although intractable sets with maximal cores do exist, this property seems to be highly unnatural. In particular, no known complete sets for NP and PSPACE are of this type. We observe that a recursive set contains a maximal core if and only if it contains a maximal P-subset, and so our results apply equally to show the nonexistence of maximal “approximations“ to natural intractable sets by P-sets.
This work was supported in part by the Emil Aaltonen Foundation, the Academy of Finland, the Deutsche Forschungsgemeinschaft, and the National Science Foundation under Grant No. MCS83-12472. The work was carried out while the first author was visiting the Department of Mathematics, University of California at Santa Barbara.
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Orponen, P., Russo, D.A., Schöning, U. (1985). Polynomial levelability and maximal complexity cores. In: Brauer, W. (eds) Automata, Languages and Programming. ICALP 1985. Lecture Notes in Computer Science, vol 194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015769
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DOI: https://doi.org/10.1007/BFb0015769
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