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.
Supported in part by the Office of the Vice Chancellor for Research and Graduate Studies at the University of Kentucky, and by the Deutsche Forschungsgemeinschaft (DFG), grant Mu 1226/2-1. Parts of the work done at University of Kentucky.
Preview
Unable to display preview. Download preview PDF.
References
V. Arvind, J. Köbler, and M. Mundhenk. On bounded truth-table, conjunctive, and randomized reductions to sparse sets. In Proceedings 12th Conference on the Foundations of Software Technology & Theoretical Computer Science, pages 140–151. Lecture Notes in Computer Science #652, Springer-Verlag, 1992.
J.L. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I/II. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1988/1990.
R. Book and J. Lutz. On languages with very high space-bounded Kolmogorov complexity. SIAM Journal on Computing, 22(2):395–402, 1993.
H. Buhrman and P. Orponen. Random strings make hard instances. In Proc. 9th Structure in Complexity Theory Conference, pages 217–222. IEEE, 1994.
L. Fortnow and M. Kummer. Resource-bounded instance complexity. In Proceedings of 12th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science #900, Springer-Verlag, 1995.
J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 439–445, 1983.
R. Karp and R. Lipton. Some relations between nonuniform and uniform complexity classes. In Prooceedings of the 12th ACM Symposium on Theory of Computing, pages 302–309, April 1980.
K. Ko. A note on the instance complexity of pseudorandom sets. In Proc. 7th Structure in Complexity Theory Conference, pages 327–337. IEEE, 1992.
K. Ko, P. Orponen, U. Schöning, and O. Watanabe. Instance complexity. Journal of the ACM, 41:96–121, 1994.
M. Kummer. The instance complexity conjecture. In Proc. 10th Structure in Complexity Theory Conference, pages 111–124. IEEE, 1995.
M. Li and P. Vitányi. An introduction to Kolmogorou complexity and its applications. Springer-Verlag, 1993.
N. Lynch. On reducibility to complex or sparse sets. Journal of the ACM, 22:341–345, 1975.
S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25(2):130–143, 1982.
P. Orponen and U. Schöning. The structure of polynomial complexity cores. In 11th Symp. on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science #176, Springer-Verlag, 1984.
O. Watanabe. Polynomial time reducibility to a set of small density. In Proc. 1987 Structure in Complexity Theory Conference, pages 138–146. Lecture Notes in Computer Science #223, Springer-Verlag, 1987.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mundhenk, M. (1997). NP-hard sets have many hard instances. In: Prívara, I., Ružička, P. (eds) Mathematical Foundations of Computer Science 1997. MFCS 1997. Lecture Notes in Computer Science, vol 1295. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0029986
Download citation
DOI: https://doi.org/10.1007/BFb0029986
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63437-9
Online ISBN: 978-3-540-69547-9
eBook Packages: Springer Book Archive