Abstract
In this paper we separate many-one reducibility from truth- table reducibility for distributional problems in Dist NP under the hy- pothesis that P ≠ NP . As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT using a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT using a less redundant distribution unless P = NP.
We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of Dist NP , this relation is proper unless P = NP. (2) C contains Dist P , but it is not contained in Ave P unless Dist NP ≠ Ave ZPP. (3) C has a ≤p m-complete set. (4) C has a ≤p tt-complete set that is not ≤p m-complete unless P = NP. This shows that under the assumption that P ≠ NP , the two complete- ness notions differ on some non-trivial subclass of Dist NP.
Supported in part by JSPS/NSF cooperative research: Complexity Theory for Strategic Goals, 1998–2001.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Aida and T. Tsukiji, On the difference among polynomial-time reducibilities for distributional problems (Japanese), in Proc. of the LA Symposium, Winter, RIMS publication, 2000.
J. Balcázar, J. Díaz, and J. Gabarró, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.
S. Ben-David, B. Chor, O. Goldreich, and M. Ludy, On the theory of average case complexity, Journal of Comput. and Syst. Sci., 44:193–219, 1992.
S.A. Cook, The complexity of theorem proving procedures, in the Proc. of the third ACM Sympos. on Theory of Comput., ACM, 151–158, 1971.
Y. Gurevich, Average case completeness, Journal of Comput. and Syst. Sci., 42:346–398, 1991.
S. Homer, Structural properties of complete problems for exponential time, in Complexity Theory Retrospective 2 (A.L. Selman Ed.), Springer-Verlag, 135–154, 1997.
R. Impagliazzo, A personal view of average-case complexity, in Proc. 10th Conference Structure in Complexity Theory, IEEE, 134–147, 1995.
S. Kirkpatrick and B. Selman, Critical Behauviour in Satisfiablility of Random Boolean Expressions, Science. 264, 1297–1301, 1994.
E. Koutsoupias and C. Papadimitriou, On the greedy algorithm for satisfiability, Infom. Process. Lett. 43, 53–55, 1992.
R. Ladner, N. Lynch, and A. Selman, A Comparison of polynomial time reducibilities, Theoretical Computer Science, 1:103–123, 1975.
L.A. Levin, Universal sequential search problem, Problems of Information Transmission, 9:265–266, 1973.
L.A. Levin, Average case completeness classes, SIAM J. Comput., 15:285–286, 1986.
L. Longpré and P. Young, Cook reducibility is faster than Karp reducibility, J. Comput. Syst. Sci., 41, 389–401, 1990.
U. Vazirani and V. Vazirani, A natural encoding scheme proved probabilistic polynomial complete, Theoret. Comput. Sci., 24, 291–300, 1983.
J. Wang, Average-case computational complexity theory, in Complexity Theory Retrospective 2 (A.L. Selman Ed.), Springer-Verlag, 295–328, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aida, S., Schuler, R., Tsukiji, T., Watanabe, O. (2001). On the Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_5
Download citation
DOI: https://doi.org/10.1007/3-540-44693-1_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41695-1
Online ISBN: 978-3-540-44693-4
eBook Packages: Springer Book Archive