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Limitations of Algebraic Approaches to Graph Isomorphism Testing

  • Christoph Berkholz
  • Martin Grohe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

Abstract

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gröbner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic \(\ne 2\) and also linear lower bounds for the degree of Positivstellensatz calculus derivations.

We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus.

Keywords

Polynomial Equation Integer Linear Program Proof System Algebraic Approach Winning Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atserias, A., Maneva, E.: Sherali-Adams relaxations and indistinguishability in counting logics. SIAM J. Comput. 42(1), 112–137 (2013)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Beame, P., Impagliazzo, R., Krajicek, J., Pitassi, T., Pudlak, P.: Lower bounds on Hilbert’s nullstellensatz and propositional proofs. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 794–806 (1994)Google Scholar
  3. 3.
    Buss, S.: Lower bounds on nullstellensatz proofs via designs. In: Proof Complexity and Feasible Arithmetics, pp. 59–71. American Mathematical Society (1998)Google Scholar
  4. 4.
    Buss, S., Grigoriev, D., Impagliazzo, R., Pitassi, T.: Linear gaps between degrees for the polynomial calculus modulo distinct primes. Journal of Computer and System Sciences 62(2), 267–289 (2001)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cai, J., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389–410 (1992)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Groebner basis algorithm to find proofs of unsatisfiability. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 174–183 (1996)Google Scholar
  7. 7.
    Codenotti, P., Schoenbeck, G., Snook, A.: Graph isomorphism and the Lasserre hierarchy (2014). CoRR arXiv:1107.0632v2
  8. 8.
    Grigoriev, D.: Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science 259(1–2), 613–622 (2001)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Grigoriev, D., Vorobjov, N.: Complexity of null- and positivstellensatz proofs. Annals of Pure and Applied Logic 113(1–3), 153–160 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Grohe, M., Otto, M.: Pebble games and linear equations. In: Cégielski, P., Durand, A. (eds.) Proceedings of the 26th International Workshop on Computer Science Logic. Leibniz International Proceedings in Informatics (LIPIcs), vol. 16, pp. 289–304 (2011)Google Scholar
  11. 11.
    Hella, L.: Logical hierarchies in PTIME. Information and Computation 129, 1–19 (1996)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11(3), 796–817 (2001)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Malkin, P.: Sherali-Adams relaxations of graph isomorphism polytopes. Discrete Optimization 12, 73–97 (2014)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    O’Donnell, R., Wright, J., Wu, C., Zhou, Y.: Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1659–1677 (2014)Google Scholar
  15. 15.
    Parrilo, P.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, California Institute of Technology (2000)Google Scholar
  16. 16.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430 (1990)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Tinhofer, G.: Graph isomorphism and theorems of Birkhoff type. Computing 36, 285–300 (1986)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Torán, J.: On the resolution complexity of graph non-isomorphism. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 52–66. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany

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