Advertisement

Combinatorica

, Volume 12, Issue 4, pp 389–410 | Cite as

An optimal lower bound on the number of variables for graph identification

  • Jin-Yi Cai
  • Martin Fürer
  • Neil Immerman
Article

Abstract

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k−1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.

AMS subject classification code (1991)

03 B 10 05 C 60 05 C 85 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. V. Aho, J. E. Hopcroft andJ. D. Ullman:The Design and Analysis of Computer Algorithms, Addison-Wesley (1974).Google Scholar
  2. [2]
    M. Ajtai: Recursive Construction for 3-Regular Expanders,28th IEEE Symp. on Foundations of Computer Science (1987), 295–304.Google Scholar
  3. [3]
    L. Babai: Monte Carlo Algorithms in Graph Isomorphism Testing, Tech. Rep. DMS 79-10, Université de Montréal, 1979.Google Scholar
  4. [4]
    L. Babai: On the Complexity of Canonical Labeling of Strongly Regular Graphs,SIAM J. Computing 9 (1980), 212–216.Google Scholar
  5. [5]
    L. Babai: Moderately Exponential Bound for Graph Isomorphism,Proc. Conf. on Fundamentals of Computation Theory, Lecture Notes in Computer Science, Springer, 1981, 34–50.Google Scholar
  6. [6]
    L. Babai: On the Order of Uniprimitive Permutation Groups,Annals of Math. 113 (1981), 553–568.Google Scholar
  7. [7]
    L. Babai:Permutation Groups, Coherent Configurations, and Graph Isomorphism, D. Sc. Thesis, Hungarian Acad. Sci., 1984 (in Hungarian).Google Scholar
  8. [8]
    L. Babai, P. Erdős, andS. M. Selkow: Random Graph Isomorphism,SIAM J. on Comput. 9 (1980), 628–635.Google Scholar
  9. [9]
    L. Babai, W. M. Kantor, andE. M. Luks: Computational Complexity and the Classification of Finite Simple Groups,24th IEEE Symp. on Foundations of Computer Science (1983), 162–171.Google Scholar
  10. [10]
    L. Babai andL. Kučera: Canonical Labelling of Graphs in Linear Average Time,20th IEEE Symp. on Foundations of Computer Science (1979), 39–46.Google Scholar
  11. [11]
    L. Babai andE. M. Luks: Canonical Labeling of Graphs,15th ACM Symposium on Theory of Computing (1983), 171–183.Google Scholar
  12. [12]
    D. M. Barrington, N. Immerman, andH. Straubing: On Uniformity Within NC1,J. Comput. System Sci. 41, No. 3 (1990), 274–306.Google Scholar
  13. [13]
    L. Babai andR. Mathon: Talk at the South-East Conference on Combinatorics and Graph Theory, 1980.Google Scholar
  14. [14]
    P. J. Cameron: 6-Transitive Graphs,J. Combinat. Theory B 28, (1980), 168–179.Google Scholar
  15. [15]
    A. Chandra andD. Harel: Structure and Complexity of Relational Queries,J. Comput. System Sci. 25 (1982), 99–128.Google Scholar
  16. [16]
    A. Ehrenfeucht: An Application of Games to the Completeness Problem for Formalized Theories,Fund. Math. 49 (1961), 129–141.Google Scholar
  17. [17]
    R. Fraïssé: Sur quelques classifications des systèms de relations,Publ. Sci. Univ. Alger 1 (1954), 35–182.Google Scholar
  18. [18]
    M. Fürer, W. Schnyder, andE. Specker: Normal Forms for Trivalent Graphs and Graphs of Bounded Valence,15th ACM Symposium on Theory of Computing (1983), 161–170.Google Scholar
  19. [19]
    Ya. Yu. Gol'fand andM. H. Klin: Onk-Regular Graphs, in:Algorithmic Research in Combinatorics, Nauka Publ., Moscow, 1978, 76–85.Google Scholar
  20. [20]
    Yu. Gurevich: Logic and the Challenge of Computer Science, in:Current Trends in Theoretical Computer Science, ed. Egon Börger, Computer Science Press, 1988, 1–57.Google Scholar
  21. [21]
    D. G. Higman: Coherent Configurations I.: Ordinary Representation Theory,Geometriae Dedicata 4 (1975), 1–32.Google Scholar
  22. [22]
    N. Immerman: Number of Quantifiers is Better than Number of Tape Cells,J. Comput. System Sci. 22, No. 3 (1981), 384–406.Google Scholar
  23. [23]
    N. Immerman: Upper and Lower Bounds for First Order Expressibility,J. Comput. System Sci. 25, No. 1 (1982), 76–98.Google Scholar
  24. [24]
    N. Immerman: Relational Queries Computable in Polynomial Time,Information and Control 68 (1986), 86–104.Google Scholar
  25. [25]
    N. Immerman: Languages That Capture Complexity Classes,SIAM J. Computing 16, No. 4 (1987), 760–778.Google Scholar
  26. [26]
    N. Immerman andD. Kozen: Definability with Bounded Number of Bound Variables,Information and Computation 83 (1989), 121–139.Google Scholar
  27. [27]
    N. Immerman andE. S. Lander: Describing Graphs: A First-Order Approach to Graph Canonization, in:Complexity Theory Retrospective, Alan Selman, ed., Springer-Verlag, 1990, 59–81.Google Scholar
  28. [28]
    N. Immerman, S. Patnaik, andD. Stemple: The Expressiveness of a Family of Finite Set Languages,Tenth ACM Symposium on Principles of Database Systems (1991), 37–52.Google Scholar
  29. [29]
    L. Kučera: Canonical Labeling of Regular Graphs in Linear Average Time,28th IEEE Symp. on Foundations of Computer Science (1987), 271–279.Google Scholar
  30. [30]
    M. H. Klin, M. E. Muzichuk, andI. A. Faradžev: Cellular Rings and Groups of Automorphism of Graphs, Introductory Article to a Book to be Published by D. Reidel Publ. Co.Google Scholar
  31. [31]
    M. Ch. Klin, R. Pöschel, andK. Rosenbaum: Angewandte Algebra, Vieweg & Sohn Publ., Braunschweig 1988.Google Scholar
  32. [32]
    R. Lipton: The Beacon Set Approach to Graph Isomorphism, Yale Dept. Comp. Sci. preprint No. 135, 1978.Google Scholar
  33. [33]
    E. M. Luks: Isomorphism of Graphs of Bounded Valence Can be Tested in Polynomial Time,J. Comput. System Sci. 25 (1982), 42–65.Google Scholar
  34. [34]
    R. Mathon: A Note On the Graph Isomorphism Counting Problem,Inform. Proc. Let. 8 (1979), 131–132.Google Scholar
  35. [35]
    B. D. McKay: Nauty User's Guide (Version 1.2), Tech. Rep. TR-CS-87-03, Dept. Computer Science, Austral. National Univ., Melbourne, 1987.Google Scholar
  36. [36]
    G. L. Miller: On then logn Isomorphism Technique,10th ACM Symposium on Theory of Computing (1978), 51–58.Google Scholar
  37. [37]
    R. C. Read andD. G. Corneil: The Graph Isomorphism Disease,J. Graph Theory 1 (1977), 339–363.Google Scholar
  38. [38]
    M. Vardi: Complexity of Relational Query Languages,14th ACM Symposium on Theory of Computing (1982), 137–146.Google Scholar
  39. [39]
    B. Weisfeiler, ed.:On Construction and Identification of Graphs, Lecture Notes in Mathematics 558, Springer, 1976.Google Scholar
  40. [40]
    B. Weisfeiler andA. A. Lehman: A Reduction of a Graph to a Canonical Form and an Algebra Arising during this Reduction, (in Russian),Nauchno-Technicheskaya Informatsia, Seriya 2,9 (1968), 12–16.Google Scholar
  41. [41]
    V. N. Zemlyachenko, N. Kornienko, andR. I. Tyshkevich:Graph Isomorphism Problem, (in Russian), The Theory fo Computation I, Notes Sci. Sem. LOMI 118, 1982.Google Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Martin Fürer
    • 2
  • Neil Immerman
    • 3
  1. 1.Computer Science Dept.Princeton UniversityPrinceton
  2. 2.Computer Science Dept.University of MassachusettsAmherst
  3. 3.Computer Science Dept.Pennsylvania State UniversityUniversity Park

Personalised recommendations