Skip to main content
SpringerLink
Log in
Book cover

Annual Symposium on Theoretical Aspects of Computer Science

STACS 1992: STACS 92 pp 401–411Cite as

Graph isomorphism is low for PP

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 577))

Abstract

We show that the graph isomorphism problem is low for PP and for C=P, i.e. it does not provide a PP or C=P computation with any additional power when used as oracle. Furthermore, we show that graph isomorphism belongs to the class LWPP (see Fenner, Fortnow, Kurtz [FeFoKu 91]). A similar result holds for the (apparently more difficult) problem Group Factorization. The problem of determining whether a given graph has a nontrivial automorphism, Graph Automorphism, is shown to be in SPP, and is therefore low for PP, C=P, and ModkP, k≥2.

This research was supported by the DAAD (Acciones Integradas 1991, 313-AI-e-es/zk). A full verstion of this paper is available as Ulmer Informatik-Bericht Nr. 91-04.

Research partially supported by ESPRIT-II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM)

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Babai, Moderately exponential bound for graph isomorphism. In Proceedings Fundamentals of Computation Theory, Lecture Notes in Computer Science 117 (1981), 34–50.

    Google Scholar 

  2. L. Babai, S. Moran, Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. In Journal of Computer and System Sciences 36 (1988), 254–276.

    Google Scholar 

  3. J.L. Balcázar, J. Díaz, J. Gabarró, Structural Complexity I. Springer, 1987.

    Google Scholar 

  4. R. Beigel, J. Gill, U. Hertrampf, Counting classes: Thresholds, parity, mods, and fewness. In Proceedings 7th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 415 (1990), 49–57.

    Google Scholar 

  5. R. Beigel, L. Hemachandra, G. Wechsung, On the power of probabilistic polynomial time: PNP[log] ⊂-PP. In Proceedings 4th Structure in Complexity Theory Conference, p. 225–227, IEEE Computer Society, 1989.

    Google Scholar 

  6. R. Boppana, J. Hastad, and S. Zachos, Does co-NP have short interactive proofs? In Information Processing Letters 25 (1987), 127–132.

    Google Scholar 

  7. J. Cai, L.A. Hemachandra, On the power of parity. In Proceedings 6th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 349 (1989), 229–240.

    Google Scholar 

  8. S.A. Cook, The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on Theory of Computing 1971, 151–158.

    Google Scholar 

  9. S. Even, A. Selman Y. Yacobi, The complexity of promise problems with applications to public-key cryptography. In Information and Control 61 (1984), 114–133.

    Google Scholar 

  10. S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes. In Proceedings of the 6th Structure in Complexity Theory Conference 1991, 30–42.

    Google Scholar 

  11. M. Furst, J. Hopcroft, E. Luks, Polynomial time algorithms for permutation groups. In Proceedings of the 21st ACM Symposium on Theory of Computing 1980, 36–41.

    Google Scholar 

  12. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.

    Google Scholar 

  13. J. Gill, Computational complexity of probabilistic Turing machines. In SIAM Journal on Computing 6 (1977), 675–695.

    Google Scholar 

  14. O. Goldreich, S. Micali, and A. Wigderson, Proofs that yield nothing but their validity and a methodology of cryptographic protocol design. In Proceedings of the 27th Symposium on Foundations of Computer Science 1986, 174–187.

    Google Scholar 

  15. S. Goldwasser, S. Micali, and C. Rackoff, The knowledge complexity of interactive proofs. In Proceedings of the 17th ACM Symposium on the Theory of Computing 1985, 291–304.

    Google Scholar 

  16. S. Goldwasser, M. Sipser, Private coins versus public coins in interactive proof systems. In Proceedings of the 18th ACM Symposium on the Theory of Computing 1986, 59–68.

    Google Scholar 

  17. F. Green, On the Power of Deterministic Reductions to C=P. Technical Report LSI-91-14 Universidad Politecnica de Catalunya, 1991.

    Google Scholar 

  18. M. Hall, The Theory of Groups, Macmillan, New York, 1959.

    Google Scholar 

  19. C. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Springer-Verlag Lecture Notes in Computer Science #136, 1982.

    Google Scholar 

  20. C. Hoffmann, Subcomplete generalizations of graph isomorphism. In Journal of Computer and System Sciences 25 (1982), 332–359.

    Google Scholar 

  21. D.S. Johnson, The NP-completeness column: An ongoing guide. In Journal of Algorithms 6 (1985), 434–451.

    Google Scholar 

  22. D. Kratsch, L.A. Hemachandra, On the complexity of graph reconstruction. In Fundamentals of Computing Theory 1991, to appear.

    Google Scholar 

  23. J. Köbler, U. Schöning, J. Torán and S. Toda, Turing Machines with few accepting computations and low sets for PP. In Proceedings of the 4th Structure in Complexity Theory Conference 1989, 208–216.

    Google Scholar 

  24. E. Luks, Isomorphism of Graphs of Bounded Valence can be tested in Polynomial Time. In Journal of Computer and System Sciences 25 (1982), 42–65.

    Google Scholar 

  25. R. Mathon, A note on the graph isomorphism counting problem. In Inform. Process. Lett. 8 (1979), 131–132.

    Google Scholar 

  26. M. Ogiwara, L. Hemachandra, A complexity theory for closure properties. In Proceedings of the 6th Structure in Complexity Theory Conference 1991, 16–29.

    Google Scholar 

  27. C. Papadimitriou, S. Zachos Two remarks on the power of counting. In 6th GI Conference on Theoretical Computer Science, Lecture Notes in Computer Science 145 (1983) 269–276.

    Google Scholar 

  28. U. Schöning, Complexity and Structure. Springer-Verlag Lecture Notes in Computer Science 211, 1986.

    Google Scholar 

  29. U. Schöning, Graph isomorphism is in the low hierarchy. In Journal of Computer and System Sciences 37 (1988), 312–323.

    Google Scholar 

  30. J. Simon, On some central problems in computational complexity. Ph.D. Thesis, Cornell University (1975).

    Google Scholar 

  31. C. Sims, Computation with permutation groups. In Proceedings of the 2nd ACM Symposium on Symbolic and Algebraic Manipulations 1971, 23–28.

    Google Scholar 

  32. J. Tarui, Degree complexity of boolean functions and its applications to relativized separations. In Proceedings of the 6th Structure in Complexity Theory Conference 1991, 382–390.

    Google Scholar 

  33. S. Toda, On the computational power of PP and ⊕P. In Proceedings of the 30th Symposium on Foundations of Computer Science 1989, 514–519.

    Google Scholar 

  34. S. Toda, Private communication.

    Google Scholar 

  35. J. Torán, An oracle characterization of the Counting Hierarchy. In Proceedings of the 3rd Structure in Complexity Theory Conference 1988, 213–224.

    Google Scholar 

  36. L.G. Valiant, The relative complexity of checking and evaluating. In Information Processing Letters 5 (1976), 20–23.

    Google Scholar 

  37. L.G. Valiant, The complexity of computing the permanent. In Theoretical Computer Science 8 (1979), 189–201.

    Google Scholar 

  38. L.G. Valiant V.V Vazirani, NP is as easy as detecting unique solutions In Theoretical Computer Science 47 (1986), 85–93.

    Google Scholar 

  39. K.W. Wagner, The complexity of combinatorial problems with succinct input representation. In Acta Informatica 23 (1986), 325–356.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretische Informatik, Universität Ulm, Oberer Eselsberg, D-7900, Ulm, Germany

    Johannes Köbler & Uwe Schöning

  2. Departamento L.S.I., U. Politecnica de Catalunya, Pau Gargallo 5, E-08028, Barcelona, Spain

    Jacobo Torán

Authors
  1. Johannes Köbler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Schöning
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jacobo Torán
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Alain Finkel Matthias Jantzen

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Köbler, J., Schöning, U., Torán, J. (1992). Graph isomorphism is low for PP. In: Finkel, A., Jantzen, M. (eds) STACS 92. STACS 1992. Lecture Notes in Computer Science, vol 577. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55210-3_200

Download citation

  • DOI: https://doi.org/10.1007/3-540-55210-3_200

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55210-9

  • Online ISBN: 978-3-540-46775-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation