Detecting almost symmetries of graphs

Full Length Paper


We present a branch-and-bound framework to solve the following problem: Given a graph G and an integer k, find a subgraph of G formed by removing no more than k edges that minimizes the number of vertex orbits. We call the symmetries on such a subgraph “almost symmetries” of G. We implement our branch-and-bound framework in PEBBL to allow for parallel enumeration and demonstrate good scaling up to 16 cores. We show that the presented branching strategy is much better than a random branching strategy on the tested graphs. Finally, we consider the presented strategy as a heuristic for quickly finding almost symmetries of a graph G. The software that was reviewed as part of this submission has been issued the Digital Object Identifier DOI:10.5281/zenodo.840558.


Almost symmetries Graph automorphism Branch-and-bound 

Mathematics Subject Classification

05C60 05C85 90C27 


  1. 1.
    Arvind, V., Köbler, J., Kuhnert, S., Vasudev, Y.: Approximate graph isomorphism. In: Mathematical Foundations of Computer Science 2012, pp. 100–111. Springer (2012)Google Scholar
  2. 2.
    Babai, L.: Graph isomorphism in quasipolynomial time. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pp. 684–697, New York, NY, USA. ACM (2016)Google Scholar
  3. 3.
    Buchheim, C., Jünger, M.: An integer programming approach to fuzzy symmetry detection. In: International Symposium on Graph Drawing, pp. 166–177. Springer (2003)Google Scholar
  4. 4.
    Culberson, J., Johnson, D., Lewandowski, G., Trick, M.: Graph coloring instances., Mar 2015
  5. 5.
    Darga, P.T., Sakallah, K.A., Markov, I.L.: Faster symmetry discovery using sparsity of symmetries. In: Proceedings of the 45th Annual Design Automation Conference, DAC ’08, pp. 149–154, New York, NY, USA. ACM (2008)Google Scholar
  6. 6.
    Eckstein, J., Hart, W.E., Phillips, C.A.: PEBBL: an object-oriented framework for scalable parallel branch and bound. Math. Program. Comput. 7(4), 429–469 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Hung. 14(3), 295–315 (1963)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pp. 534–543. ACM (2002)Google Scholar
  9. 9.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. In: Proceedings of the Eleventh Annual IEEE Conference on Computational Complexity, 1996, pp. 278–287. IEEE (1996)Google Scholar
  10. 10.
    Fox, M., Long, D., Porteous, J.: Discovering near symmetry in graphs. In: Proceedings of the 22nd National Conference on Artificial Intelligence, vol. 1, pp. 415–420. AAAI Press (2007)Google Scholar
  11. 11.
    Fürstenberg, C.: A drawing of a graph., Mar (2015)
  12. 12.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete problems. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 47–63. ACM (1974)Google Scholar
  13. 13.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in np have zero-knowledge proof systems. J ACM 38(3), 690–728 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Knuth, D.E.: The Stanford GraphBase: A Platform for Combinatorial Computing, vol. 37. Addison-Wesley, Reading (1993)MATHGoogle Scholar
  15. 15.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Log. Q 2(1–2), 83–97 (1955)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lin, C.-L.: Hardness of approximating graph transformation problem. In: Algorithms and Computation, pp. 74–82. Springer (1994)Google Scholar
  17. 17.
    Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program. 94(1), 71–90 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Margot, F.: Exploiting orbits in symmetric ILP. Math. Program. 98(1–3), 3–21 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Markov, I.: Almost-symmetries of graphs. In: Proceedings of International Symmetry Conference (ISC), pp. 60–70 (2007)Google Scholar
  20. 20.
    Mathon, R.: A note on the graph isomorphism counting problem. Inf. Process. Lett. 8(3), 131–136 (1979)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    McKay, B.D.: Practical Graph Isomorphism. Department of Computer Science, Vanderbilt University (1981)Google Scholar
  22. 22.
    McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    McKay, B.D., Piperno, A.: Nauty traces—graphs., Mar (2015)
  24. 24.
    Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    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 Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1659–1677. SIAM (2014)Google Scholar
  26. 26.
    Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126(1), 147–178 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Read, R.C., Corneil, D.G.: The graph isomorphism disease. J. Gr. Theory 1(4), 339–363 (1977)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Stachniss, C.: C implementation of the Hungarian method., Mar (2015)

Copyright information

© Springer-Verlag GmbH Germany and The Mathematical Programming Society 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of TennesseeKnoxvilleUSA
  2. 2.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations