Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T12:06:27.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Restrictive Acceptance Suffices for Equivalence Problems

Published online by Cambridge University Press:  01 February 2010

Bernd Borchert ,
Lane A. Hemaspaandra  and
Jörg Rothe
Bernd Borchert
Affiliation:
Mathematisches Institut, Universität Heidelberg, 69120 Heidelberg, Germany, [email protected]
Lane A. Hemaspaandra
Affiliation:
Department of Computer Science, University of Rochester, Rochester, NY 14627, USA, [email protected]
Jörg Rothe
Affiliation:
Institut für Informatik, Friedrich-Schiller-Universität Jena, 07740 Jena, Germany, [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One way of suggesting that an NP problem may not be NP-complete is to show that it is in the promise class UP. We propose an analogous new method—weaker in strength of evidence but more broadly applicable—for suggesting that concrete NP problems are not NP-complete. In particular, we introduce the promise class EP, the subclass of NP consisting of those languages accepted by NP machines that, when they accept, always have a number of accepting paths that is a power of two. We show that FewP, bounded ambiguity polynomial time (which contains UP), is contained in EP. The class EP applies as an upper bound to some concrete problems to which previous approaches have never been successful, for example the negation equivalence problem for OBDDs (ordered binary decision diagrams).

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 86 - 95
Copyright
Copyright © London Mathematical Society 2000

References

1. Agrawal, M. and Thierauf, T., ‘The boolean isomorphism problem’, Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (IEEE Computer Society Press, 1996) 422–430.Google Scholar
2. Allender, E. and Rubinstein, R., ‘P-printable sets’, SIAM J. Comput. 17 (1988) 1193–1202.CrossRefGoogle Scholar
3. Beigel, R., ‘On the relativized power of additional accepting paths’, Proceedings of the 4th Structure in Complexity Theory Conference (IEEE Computer Society Press, 1989)216–224.Google Scholar
4. Beigel, R., Chang, R. and Ogiwara, M., ‘A relationship between difference hierarchies and relativized polynomial hierarchies’, Math. Systems Theory 26 (1993) 293–310.CrossRefGoogle Scholar
5. Beigel, R., Gill, J. and Hertrampf, U., ‘Counting classes: thresholds, parity, mods, and fewness’, Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 415 (ed. Choffrut, C. and Lengauer, T., Springer-Verlag, 1990) 49–57.Google Scholar
6. Blass, A. and Gurevich, Y., ‘On the unique satisfiability problem’, Information and Control 55 (1982) 80–88.CrossRefGoogle Scholar
7. Borchert, B. and Stephan, F., ‘Looking for an analogue of Rice's theorem in circuit complexity theory’, Math. Logic Quart. To appear.Google Scholar
8. Borchert, B., Hemaspaandra, L. and Rothe, J., ‘Powers-of-two acceptance suffices for equivalence and bounded ambiguity problems’, Tech. Rep. TR96–045, Electronic Colloquium on Computational Complexity, August 1996 http://www.eccc.uni-trier.de/eccc/.Google Scholar
9. Borchert, B., Hemaspaandra, L. and Rothe, J., ‘Restrictive acceptance suffices for equivalence problems’, Proceedings of the 12th Conference on Fundamentals of Computation Theory, Lecture Notes in Comput. Sci. 1684 (ed. Ciobanu, G. and Paun, G., Springer Verlag, 1999) 124–135.Google Scholar
10. Borchert, B., Ranjan, D. and Stephan, F., ‘On the computational complexity of some classical equivalence relations on boolean functions’, Theory Comput. Syst. 31 (1998) 679–693.CrossRefGoogle Scholar
11. Bryant, R., ‘Symbolic boolean manipulation with ordered binary decision diagrams’, ACM Computing Surveys 24 (1992) 293–318.CrossRefGoogle Scholar
12. Cai, J. and Hemachandra, L., ‘On the power of parity polynomial time’, Math. Systems Theory 23 (1990) 95–106.CrossRefGoogle Scholar
13. Feigenbaum, J., Kannan, S., Vardi, M. and Vishvanathan, M., ‘Complexity of problems on graphs represented as OBDDs’, Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1373 (ed. Morvan, M., Meinel, C. and Krob, D., Springer-Verlag, 1998) 216–226.Google Scholar
14. Fenner, S., Fortnow, L. and Kurtz, S., ‘Gap-definable counting classes’, J. Comput. System Sci. 48 (1994) 116–148.CrossRefGoogle Scholar
15. Fortune, S., Hopcroft, J. and Schmidt, E., ‘The complexity of equivalence and containment for free single program schemes’, Proceedings of the 5th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 62 (ed. Ausiello, G. and Böhm, C., Springer-Verlag, 1978) 227–240.Google Scholar
16. Grollmann, J. and Selman, A., ‘Complexity measures for public-key cryptosystems’, SIAMJ. Comput. 17 (1988) 309–335.CrossRefGoogle Scholar
17. Harrison, M., ‘Counting theorems and their applications to classification of switching functions’, Recent developments in switching theory, (ed. Mukhopadyay, A., Academic Press, 1971)4–22.Google Scholar
18. Hartmanis, J. and Yesha, Y., ‘Computation times of NP sets of different densities’, Theoret. Comput. Sci. 34 (1984) 17–32.CrossRefGoogle Scholar
19. Hemaspaandra, L. and Rothe, J., ‘Unambiguous computation: Boolean hierarchies and sparse Turing-complete sets’, SIAMJ. Comput. 26 (1997) 634–653.CrossRefGoogle Scholar
20. Hemaspaandra, L. and Rothe, J., ‘A second step towards complexity-theoretic analogs of Rice's theorem’, Theoret. Comput. Sci To appear.Google Scholar
21. Hemaspaandra, L., Jain, S. and Vereshchagin, N., ‘Banishing robust Turing completeness’, Internat. J. Found. Comput. Sci 4 (1993) 245–265.CrossRefGoogle Scholar
22. Hoffmann, C., Group-theoretic algorithms and graph isomorphism, Lecture Notes in Comput. Sci. 136 (Springer-Verlag, 1982).CrossRefGoogle Scholar
23. Köbler, J., Schöning, U., Toda, S. and Torán, J., ‘Turing machines with few accepting computations and low sets for PP’, J. Comput. System Sci. 44 (1992) 272–286.CrossRefGoogle Scholar
24. Köbler, J., Schöning, U. and Torán, J., The graph isomorphism problem: its structural complexity (Birkhauser, 1993).CrossRefGoogle Scholar
25. Luks, E., ‘Isomorphism of graphs of bounded valence can be tested in polynomial time’, J. Comput. System Sci. 25 (1982) 42–65.CrossRefGoogle Scholar
26. Rao, R., Rothe, J. and Watanabe, O., ‘Upward separation for FewP and related classes’, Inform. Process. Lett. 52 (1994) 175–180.CrossRefGoogle Scholar
27. Schöning, U., ‘Probabilistic complexity classes and lowness’, J. Comput. System Sci. 39 (1989) 84–100.CrossRefGoogle Scholar
28. Takenaga, Y., Nouzoe, M. and Yajima, S., ‘Size and variable ordering of OBDDs representing threshold functions’, Proceedings of the 3rd Annual International Computing and Combinatorics Conference, Lecture Notes in Comput. Sci. 1276 (ed.Jiang, T. and Lee, D. T., Springer-Verlag, 1997) 91–100.Google Scholar
29. Valiant, L., ‘The relative complexity of checking and evaluating’, Inform. Process. Lett. 5 (1976)20–23.CrossRefGoogle Scholar
30. Wagner, K., ‘The complexity of combinatorial problems with succinct input representations’, Acta Inform. 23 (1986) 325–356.CrossRefGoogle Scholar