, Volume 67, Issue 2, pp 247–276 | Cite as

The Parallel Complexity of Graph Canonization Under Abelian Group Action

  • V. ArvindEmail author
  • Johannes Köbler


We study the problem of computing canonical forms for graphs and hypergraphs under Abelian group action and show tight complexity bounds. Our approach is algebraic. We transform the problem of computing canonical forms for graphs to the problem of computing canonical forms for associated algebraic structures, and we develop parallel algorithms for these associated problems.
  1. 1.

    In our first result we show that the problem of computing canonical labelings for hypergraphs of color class size 2 is logspace Turing equivalent to solving a system of linear equations over the field \(\mathbb {F} _{2}\). This implies a deterministic NC 2 algorithm for the problem.

  2. 2.

    Similarly, we show that the problem of canonical labeling graphs and hypergraphs under arbitrary Abelian permutation group action is fairly well characterized by the problem of computing the integer determinant. In particular, this yields deterministic NC 3 and randomized NC 2 algorithms for the problem.



Graph isomorphism Logspace counting classes Parallel complexity 


  1. 1.
    Allender, E.: Arithmetic circuits and counting complexity classes. In: Krajíček, J. (ed.) Complexity of Computations and Proofs. Quaderni di Matematica, vol. 13, pp. 33–72. Seconda Universita di Napoli, Naples (2004) Google Scholar
  2. 2.
    Allender, E., Ogihara, M.: Relationships among PL, #L and the determinant. RAIRO Theor. Inform. Appl. 30(1), 1–21 (1996) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Allender, E., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Comput. Complex. 8, 99–126 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arvind, V., Köbler, J.: On hypergraph and graph isomorphism with bounded color classes. In: Proceedings of 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 3884, pp. 384–395. Springer, Berlin (2006) Google Scholar
  5. 5.
    Arvind, V., Vijayaraghavan, T.C.: The complexity of solving linear equations over a finite ring. In: Proceedings 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 3404, pp. 472–484. Springer, Berlin (2005) Google Scholar
  6. 6.
    Arvind, V., Das, B., Mukhopadhyay, P.: Isomorphism and canonization of tournaments and hypertournaments. J. Comput. Syst. Sci. 76(7), 509–523 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: Proceedings 15th Annual ACM Symposium on Theory of Computing (STOC), pp. 171–183 (1983) Google Scholar
  8. 8.
    Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace-MOD class. Theory Comput. Syst. 25(3), 223–237 (1992) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Buss, S.R.: Alogtime algorithms for tree isomorphism, comparison, and canonization. In: Computational Logic and Proof Theory. Lecture Notes in Computer Science, vol. 1289, pp. 18–33. Springer, Berlin (1997) CrossRefGoogle Scholar
  10. 10.
    Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Inf. Control 64(1–3), 2–22 (1985) zbMATHCrossRefGoogle Scholar
  11. 11.
    Damm, C.: DET=L #l. Technical Report Informatik-Preprint 8, Fachbereich Informatik der Humboldt-Universität zu Berlin (1991) Google Scholar
  12. 12.
    Datta, S., Limaye, N., Nimbhorkar, P., Thierauf, T., Wagner, F.: Planar graph isomorphism is in log-space. In: Proceedings 24th Annual IEEE Conference on Computational Complexity (CCC), pp. 203–214. IEEE Comput. Soc., Los Alamitos (2009) Google Scholar
  13. 13.
    Furst, M., Hopcroft, J.E., Luks, E.M.: Polynomial-time algorithms for permutation groups. Technical report, Cornell University (1980) Google Scholar
  14. 14.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. J. Comput. Syst. Sci. 66(3), 549–566 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Karchmer, M., Wigderson, A.: On span programs. In: Proc. of the 8th IEEE Structure in Complexity Theory Conference, pp. 102–111. IEEE Comput. Soc., Los Alamitos (1993) Google Scholar
  16. 16.
    Köbler, J., Schöning, U., Torán, J.: Graph Isomorphism: Its Structural Complexity. Birkhäuser, Boston (1992) Google Scholar
  17. 17.
    Lindell, S.: A logspace algorithm for tree canonization. In: Proceedings 24th Annual ACM Symposium on Theory of Computing (STOC), pp. 400–404 (1992) Google Scholar
  18. 18.
    Luks, E.M.: Parallel algorithms for permutation groups and graph isomorphism. In: Proceedings 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, pp. 292–302 (1986) Google Scholar
  19. 19.
    Luks, E.M.: Permutation groups and polynomial-time computation. In: Finkelstein, L., Kantor, W.M. (eds.) Groups and Computation. Discrete Mathematics and Theoretical Computer Science, vol. 11, pp. 139–175. Am. Math. Soc., Providence (1993) Google Scholar
  20. 20.
    Luks, E.M., McKenzie, P.: Parallel algorithms for solvable permutation groups. J. Comput. Syst. Sci. 37(1), 39–62 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McKenzie, P., Cook, S.A.: The parallel complexity of Abelian permutation group problems. SIAM J. Comput. 16(3), 880–909 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Miller, G.L., Reif, J.H.: Parallel tree contraction part 2: further applications. SIAM J. Comput. 20(6), 1128–1147 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Reingold, O.: Undirected st-connectivity in log-space. In: Proceedings 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 376–385 (2005) Google Scholar
  24. 24.
    Toda, S.: Counting problems computationally equivalent to the determinant. Manuscript (1991) Google Scholar
  25. 25.
    Torán, J.: On the hardness of graph isomorphism. SIAM J. Comput. 33(5), 1093–1108 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Valiant, L.G.: Why is boolean complexity theory difficult? In: Hemaspaandra, L.A., Selman, A.L. (eds.) Boolean Function Complexity. Lecture Notes Series, vol. 169, pp. 53–80. Cambridge University Press, Cambridge (1992). London Mathematical Society Google Scholar
  27. 27.
    Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Proc. of the 6th IEEE Structure in Complexity Theory Conference, pp. 270–284 (1991) Google Scholar
  28. 28.
    Wielandt, H.: Permutation Groups. Academic Press, New York (1964) zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesC. I. T. CampusChennaiIndia
  2. 2.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany

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