On the Resolution Complexity of Graph Non-isomorphism

  • Jacobo Torán
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7962)


For a pair of given graphs we encode the isomorphism principle in the natural way as a CNF formula of polynomial size in the number of vertices, which is satisfiable if and only if the graphs are isomorphic. Using the CFI graphs from [12], we can transform any undirected graph G into a pair of non-isomorphic graphs. We prove that the resolution width of any refutation of the formula stating that these graphs are isomorphic has a lower bound related to the expansion properties of G. Using this fact, we provide an explicit family of non-isomorphic graph pairs for which any resolution refutation requires an exponential number of clauses in the size of the initial formula. These graphs pairs are colored with color multiplicity bounded by 4. In contrast we show that when the color classes are restricted to have size 3 or less, the non-isomorphism formulas have tree-like resolution refutations of polynomial size.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ajtai, M.: Recursive construction for 3-regular expanders. Combinatorica 14(4), 379–416 (1994)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. Computational Complexity 20(4), 649–678 (2011)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Anton, C., Neal, C.: Notes on generating satisfiable SAT instances using random subgraph isomorphism. In: Farzindar, A., Kešelj, V. (eds.) Canadian AI 2010. LNCS, vol. 6085, pp. 315–318. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Anton, C.: An improved satisfiable SAT generator based on random subgraph isomorphism. In: Butz, C., Lingras, P. (eds.) Canadian AI 2011. LNCS, vol. 6657, pp. 44–49. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Arvind, V., Kurur, P.P., Vijayaraghavan, T.C.: Bounded color multiplicity graph isomorphism is in the #L Hierarchy. In: Proceedings of the 20th Conference on Computational Complexity, pp. 13–27 (2005)Google Scholar
  6. 6.
    Babai, L.: Monte Carlo algorithms for Graph Isomorphism testing. Tech. Rep. 79-10, Dép. Math. et Stat., Univ. de Montréal (1979)Google Scholar
  7. 7.
    Beame, P., Culberson, J.C., Mitchell, D.G., Moore, C.: The resolution complexity of random graph k-colorability. Discrete Applied Mathematics 153(1-3), 25–47 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Beame, P., Impagliazzo, R., Sabharwal, A.: The resolution complexity of independent sets and vertex covers in random graphs. Computational Complexity 16(3), 245–297 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Beame, P., Pitassi, T.: Simplified and improved resolution lower bounds. In: 37th Annual IEEE Symposium on Foundations of Computer Science, pp. 274–282 (1996)Google Scholar
  10. 10.
    Ben-Sasson, E., Impagliazzo, R., Wigderson, A.: Near-optimal separation of treelike and general resolution. Combinatorica 24(4), 585–603 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow – resolution made simple. Journal of the ACM 48(2), 149–169 (2001)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Cai, J., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identifications. Combinatorica 12(4), 389–410 (1992)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chvátal, V., Szemerédi, E.: Many hard examples for resolution. Journal of the ACM 35, 759–768 (1988)MATHCrossRefGoogle Scholar
  14. 14.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5, 394–397 (1962)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Furst, M., Hopcroft, J., Luks, E.: Polynomial time algorithms for permutation groups. In: Proc. 21st IEEE Symp. on Foundations of Computer Science, pp. 36–41 (1980)Google Scholar
  16. 16.
    Goerdt, A.: The cutting plane proof system with bounded degree of falsity. In: Kleine Büning, H., Jäger, G., Börger, E., Richter, M.M. (eds.) CSL 1991. LNCS, vol. 626, pp. 119–133. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  17. 17.
    Haken, A.: The intractability of resolution. Theoretical Computer Science 39(2-3), 297–308 (1985)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Hirsch, E.A., Kojevnikov, A., Kulikov, A.S., Nikolenko, S.I.: Complexity of semialgebraic proofs with restricted degree of falsity. Journal on Satisfiability, Boolean Modeling and Computation 6, 53–69 (2008)MathSciNetGoogle Scholar
  19. 19.
    Immerman, N., Lander, E.: Describing graphs: a first-order approach to graph canonization. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer (1990)Google Scholar
  20. 20.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. J. Comput. Syst. Sci. 66(3), 549–566 (2003)MATHCrossRefGoogle Scholar
  21. 21.
    Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism problem: Its structural complexity. Birkhauser (1993)Google Scholar
  22. 22.
    Robinson, J.A.: A machine oriented logic based on the resolution principle. Journal of the ACM 12(1), 23–41 (1965)MATHCrossRefGoogle Scholar
  23. 23.
    Schöning, U., Torán, J.: Das Erfüllbarkeitsproblem SAT - Algorithmen und Analysen, Lehmann (2012)Google Scholar
  24. 24.
    Torán, J.: On the hardness of Graph Isomorphism. SIAM Journal on Computing 33(5), 1093–1108 (2004)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic, Part 2, pp. 115–125. Consultants Bureau (1968)Google Scholar
  26. 26.
    Urquhart, A.: Hard examples for resolution. Journal of the ACM 34, 209–219 (1987)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jacobo Torán
    • 1
  1. 1.Institut für Theoretische InformatikUniversität UlmUlmGermany

Personalised recommendations