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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

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Generic case complexity of the Graph Isomorphism Problem

Gennady A. Noskov / Alexander N. Rybalov
  • Corresponding author
  • Omsk State Technical University, Omsk, Russia; and Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk, Russia
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Published Online: 2016-04-09 | DOI: https://doi.org/10.1515/gcc-2016-0008


The edge test is a partial algorithm for the Graph Isomorphism Problem based on comparison the number of edges. We perform a probabilistic analysis of the efficiency of the edge test. With the binomial distribution B(n,p) on the set of inputs, we estimate the asymptotic failure probability of the edge test depending on the rate of decay of parameter p. In particular, if p ≤ 1/2, np → λ > 0, then the asymptotic failure probability is nonzero, so that the edge test does not solve generically the Graph Isomorphism Problem. On the other hand, if p ≤ 1/2, np → ∞, then the failure set is negligible and the edge test generically solves the Graph Isomorphism Problem in polynomial time.

Keywords: Graph isomorphism; binomial random graph; Poisson distribution; Legendre polynomial; generic algorithm

MSC: 05C38; 15A15; 05A15; 15A18


  • 1

    L. Babai, P. Erdős and S. M. Selkow, Random graph isomorphism, SIAM J. Comput. 9 (1980), 3, 628–635. Google Scholar

  • 2

    A. Bogdanov and L. Trevisan, Average-case Complexity, Found. Trends Theor. Comput. Sci. 2 (2006), 1, 1–111. Google Scholar

  • 3

    B. Bollobas, Random Graphs, Academic Press, London, 1985. Google Scholar

  • 4

    J. D. Burtin, Extremal metric characteristics of a random graph. II: Limit distributions (in Russian), Teor. Verojatnost. i Primenen 20 (1975), 82–99. Google Scholar

  • 5

    P. J. Davis, Interpolation and Approximation, 2nd ed., Dover Publications, New York, 1975. Google Scholar

  • 6

    G. M. Fichtengolc, A Course of Differential and Integral Calculus II (in Russian), Nauka, Moscow, 1970. Google Scholar

  • 7

    A. M. Frieze and B. Reed, Probabilistic analysis of algorithms, Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms Combin. 16, Springer, Berlin (1998), 36–92. Google Scholar

  • 8

    R. Gilman, A. G. Miasnikov, A. D. Myasnikov and A. Ushakov, Report on generic case complexity, Herald Omsk Univ. (2007), Spec. Iss., 103–110. Google Scholar

  • 9

    H. Haber, The normal approximation to the binomial distribution, handout, University of California, Santa Cruz, 2012, http://scipp.ucsc.edu/~haber/ph116C/NormalApprox.pdf.

  • 10

    E. Heine, Handbuch der Kugelfunctionen, Vol. I, 2nd ed., G. Reimer, Berlin, 1878. Google Scholar

  • 11

    E. Heine, Handbuch der Kugelfunctionen, Vol. II, 2nd ed., G. Reimer, Berlin, 1881. Google Scholar

  • 12

    C. M. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Comput. Sci. 136, Springer, Berlin, 1982. Google Scholar

  • 13

    R. Impagliazzo, A personal view of average-case complexity, Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT 1995), IEEE Press, Piscateway (1995), 134–147. Google Scholar

  • 14

    R. Karp, The fast approximate solution of hard combinatorial problems, Proceedings of the 6th southeastern conference on combinatorics, graph theory, and computing (Boca Raton 1975), Congr. Numer. 14, Utilitas Mathematica Publishing, Winnipeg (1975), 15–31. Google Scholar

  • 15

    A. D. Korshunov, The main properties of random graphs with a large number of vertices and edges, Uspekhi Mat. Nauk 40(241) (1985), 1, 107–173. Google Scholar

  • 16

    R. Lipton, The beacon set approach to graph isomorphism, Research Report 135, Yale University, New Haven, 1978. Google Scholar

  • 17

    I. I. Lyashko, A. K. Boyarchuk, Y. G. Gaĭ and A. F. Kalaĭda, Mathematical Analysis. Part 1 (in Russian), Editorial URSS, Moskva, 2001. Google Scholar

  • 18

    A. Myasnikov and A. Rybalov, Generic complexity of undecidable problems, J. Symb. Log. 73 (2008), 2, 656–673. Web of ScienceGoogle Scholar

  • 19

    G. Polya and G. Szegő, Problems and Theorems in Analysis. I. Series, Integral Calculus, Theory of Functions, Classics Math., Springer, Berlin, 1998. Google Scholar

  • 20

    G. Polya and G. Szegő, Problems and Theorems ín Analysis. II. Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry, Classics Math., Springer, Berlin, 1998. Google Scholar

  • 21

    Y. V. Prohorov, Asymptotic behavior of the binomial distribution (in Russian), Uspekhi Mat. Nauk 8 (1953), 3(55), 135–142. Google Scholar

  • 22

    R. C. Read and D. G. Corneil, The graph isomorphism disease, J. Graph Theory 1 (1977), 4, 339–363. Google Scholar

  • 23

    J. Schreyer, Almost every graph is vertex-oblique, Discrete Math. 307 (2007), 983–989. Web of ScienceGoogle Scholar

  • 24

    A. N. Shiryaev, Probability, 2nd ed., Grad. Texts in Math. 95, Springer, New York, 1996. Google Scholar

  • 25

    G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 33, American Mathematical Society, Providence, 1975. Google Scholar

About the article

Received: 2014-06-08

Published Online: 2016-04-09

Published in Print: 2016-05-01

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 14-01-00068

Funding Source: DFG

Award identifier / Grant number: SFB 701 of Bielefeld University

Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-11-00085

Gennady Noskov acknowledges financial support from the DFG through SFB 701 of Bielefeld University, Germany and from the Russian Foundation for Basic Research (RFFI-grant 14-01-00068), Russia. Alexander Rybalov acknowledges the Russian Science Foundation (under grant 14-11-00085) for financial support.

Citation Information: Groups Complexity Cryptology, Volume 8, Issue 1, Pages 9–20, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0008.

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