Skip to main content
SpringerLink
Log in

Graph isomorphism is low for PP

  • Published:
computational complexity Aims and scope Submit manuscript

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 an oracle. Furthermore, we show that graph isomorphism belongs to the class LWPP (see Fenner, Fortnow, Kurtz [12]). 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 Mod k P,k≥2.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

  2. L. Babai, G. Cooperman, L. Finkelstein, E. Luks, Á. Seress, Fast Monte Carlo algorithms for permutation groups, inProc. 23rd ACM Symposium on Theory of Computing, 1991, 90–100.

  3. L. Babai, E. Luks, Á. Seress, Fast management of permutation groups, inProc. 28th IEEE Symposium on Foundations of Computer Science, 1988, 272–282.

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

    Google Scholar 

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

  6. R. Beigel andJ. Gill. Counting classes: thresholds, parity, mods, and fewness,Theoretical Computer Science 103, (1992), 3–23.

    Google Scholar 

  7. R. Beigel, L. Hemachandra., G. Wechsung, Probabilistic polynomial time is closed under parity reductions,Information Processing Letters 37 (1991), 91–94.

    Google Scholar 

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

    Google Scholar 

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

  10. S.A. Cook, The complexity of theorem-proving procedures, inProc. 3rd ACM Symposium on Theory of Computing, 1971, 151–158.

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

    Google Scholar 

  12. S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, inProc. 6th Structure in Complexity Theory Conference, 1991, 30–42.

  13. M. Furst, J. Hopcroft, E. Luks, Polynomial time algorithms for permutation groups, inProc. 21st IEEE Symposium on Foundations of Computer Science, 1980, 36–41.

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

    Google Scholar 

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

    Google Scholar 

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

  17. S. Goldwasser, S. Micali, andC. Rackoff, The knowledge complexity of interactive proofs,SIAM Journal on Computing,18(1), (1989), 186–208.

    Google Scholar 

  18. S. Goldwasser, M. Sipser, Private coins versus public coins in interactive proof systems, inRandomness and Computation., S. Micali, editor, volume 5 ofAdvances in Computing Research, JAI Press, 1989, 73–90.

  19. F. Green, On the Power of Deterministic Reductions to C=P, to appear inMathematical System Theory.

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

    Google Scholar 

  21. U. Hertrampf, Relations among MOD-classes,Theoretical Computer Science 74 (1990), 325–328.

    Google Scholar 

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

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

    Google Scholar 

  24. M. Jerrum, A compact representation for permutation groups,Journal of Algorithms 7 (1986), 60–78.

    Google Scholar 

  25. D.S. Johnson, The NP-completeness column: An ongoing guideJournal of Algorithms 6 (1985), 434–451.

    Google Scholar 

  26. D. Kratsch, L.A. Hemachandra, On the complexity of graph reconstruction, inProc. 8th Fundamentals of Computation Theory Conference, Lecture Notes in Computer Science #529 (1991), 318–328. To appear inMathematical Systems Theory.

  27. J. Köbler, U. Schöning, J. Torán andS. Toda, Turing Machines with few accepting computations and low sets for PP,Journal of Computer and System Sciences 44 (1992), 272–286.

    Google Scholar 

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

    Google Scholar 

  29. R. Mathon, A note on the graph isomorphism counting problem,Information Processing Letters 8 (1979), 131–132.

    Google Scholar 

  30. M. Ogiwara, L. Hemachandra, A complexity theory for closure properties, inProc. 6th Structure in Complexity Theory Conference, 1991, 16–29.

  31. C. Papadimitriou, S. Zachos, Two remarks on the power of counting, in6th GI Conference on Theoretical Computer Science, Lecture Notes in Computer Science #145 (1983), 269–276.

  32. U. Schöning,Complexity and Structure, Lecture Notes in Computer Science #211, Springer, 1986.

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

    Google Scholar 

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

  35. C. Sims, Computation with permutation groups, inProc. 2nd ACM Symposium on Symbolic and Algebraic Manipulations, 1971, 23–28.

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

  37. S. Toda, PP is as hard as the polynomial-time hierarchy,SIAM Journal on Computing 20(5) (1991), 865–877.

    Google Scholar 

  38. S. Toda, Private communication.

  39. J. Torán, Complexity classes defined by counting quantifiers,Journal of the ACM 38 (1991), 753–774.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

Download references

Author information

Authors and Affiliations

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

    Johannes Köbler & Uwe Schöning

  2. Departamento L.S.I., Universitat Politècnica 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

Rights and permissions

Reprints and permissions

About this article

Cite this article

Köbler, J., Schöning, U. & Torán, J. Graph isomorphism is low for PP. Comput Complexity 2, 301–330 (1992). https://doi.org/10.1007/BF01200427

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01200427

Key words

Subject classifications

Search

Navigation