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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References
J.L.Balcázar and U.Schöning, Bi-immune sets for complexity classes. To appear in Math. Syst. Theory.
L.Berman, On the structure of complete sets: almost everywhere complexity and infinitely often speedup. Proc. 17th IEEE Symp. Foundations of Computer Science (1976), 76–80.
L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets. SIAM J. Comput. 6 (1977), 305–322.
M. Blum and I. Marques, On complexity properties of recursively enumerable sets. J. Symb. Logic 38 (1973), 579–593.
P.Flajolet and J.M.Steyaert, On sets having only hard subsets. Proc. 2nd Int. Colloq. Automata, Languages, and Programming (1974), Springer-Verlag, 446–457.
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, Ma., 1979.
R.Karp and R.Lipton, Some connections between nonuniform and uniform complexity classes. Proc. 12th ACM Symp. on Theory of Computing (1980), 302–309.
K.Ko, Non-levelable sets and immune sets in the accepting density hierarchy in NP. Manuscript (1984), submitted for publication.
N. Lynch, On reducibility to complex or sparse sets. J. Assoc. Comput. Mach. 22 (1975), 341–345.
S.R.Mahaney and P.Young, Orderings of polynomial isomorphism types. Manuscript (1983), submitted for publication.
A.R.Meyer and M.S.Paterson, With what frequency are apparently intractable problems difficult? Tech. Rep. TM-126, Laboratory of Computer Science, Massachusetts Institute of technology, Cambridge, Ma., 1979.
P.Orponen, A classification of complexity core lattices. Manuscript (1984), submitted for publication.
P. Orponen and U. Schöning, The structure of polynomial complexity cores. Proc. 11th Symp. Math. Foundations of Computer Science (1984), Lecture Notes in Computer Science 176, Springer-Verlag, 452–458.
D.A.Russo and P.Orponen, A duality between recursive complexity cores and polynomial time computable subsets. Manuscript (1984), submitted for publication.
C.P.Schnorr, Optimal algorithms for self-reducible problems. Proc. 3rd Int. Colloq. Automata, Languages, and Programming (1976), Edinburgh Univ. Press, 322–337.
A.L. Selman, Reductions on NP and P-selective sets. Theoret. Computer Sci. 19 (1982), 287–304.
R. Soare, Computational complexity, speedable and levelable sets. J. Symb. Logic 42 (1977), 545–563.
P.Young, Some structural properties of polynomial reducibilities. Proc. 15th ACM Symp. Theory of Computing (1983), 392–401.
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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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