An Integer Programming Approach to Fuzzy Symmetry Detection

  • Christoph Buchheim
  • Michael Jünger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)

Abstract

The problem of exact symmetry detection in general graphs has received much attention recently. In spite of its NP-hardness, two different algorithms have been presented that in general can solve this problem quickly in practice [5,2]. However, as most graphs do not admit any exact symmetry at all, the much harder problem of fuzzy symmetry detection arises: a minimal number of certain modifications of the graph should be allowed in order to make it symmetric. We present a general approach to this problem: we allow arbitrary edge deletions and edge creations; every single modification can be given an individual weight. We apply integer programming techniques to solve this problem exactly or heuristically and give runtime results for a first implementation.

References

  1. 1.
    ABACUS – A Branch-And-CUt System, http://www.informatik.uni-koeln.de/abacus
  2. 2.
    Abelson, D., Hong, S., Taylor, D.: A group-theoretic method for drawing graphs symmetrically. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 86–97. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Buchheim, C.: An Integer Programming Approach to Exact and Fuzzy Symmetry Detection. PhD thesis, Institut für Informatik, Universität zu Köln (2003), Available at kups.ub.uni-koeln.de/volltexte/2003/918
  4. 4.
    Buchheim, C., Jünger, M.: Detecting symmetries by branch & cut. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 178–188. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Buchheim, C., Jünger, M.: Detecting symmetries by branch&cut. Mathematical Programming, Series B 98, 369–384 (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Chen, H.-L., Lu, H.-I., Yen, H.-C.: On maximum symmetric subgraphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 372–383. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
  8. 8.
    Eades, P., Lin, X.: Spring algorithms and symmetry. Theoretical Computer Science 240(2), 379–405 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hong, S., Eades, P.: Drawing planar graphs symmetrically II: Biconnected graphs. Technical Report CS-IVG-2001-01, University of Sydney (2001)Google Scholar
  10. 10.
    Hong, S., Eades, P.: Drawing planar graphs symmetrically III: Oneconnected graphs. Technical Report CS-IVG-2001-02, University of Sydney (2001)Google Scholar
  11. 11.
    Hong, S., Eades, P.: Drawing planar graphs symmetrically IV: Disconnected graphs. Technical Report CS-IVG-2001-03, University of Sydney (2001)Google Scholar
  12. 12.
    Hong, S., McKay, B., Eades, P.: Symmetric drawings of triconnected planar graphs. In: SODA 2002, pp. 356–365 (2002)Google Scholar
  13. 13.
    Manning, J.: Computational complexity of geometric symmetry detection in graphs. In: Sherwani, N.A., Kapenga, J.A., de Doncker, E. (eds.) Great Lakes CS Conference 1989. LNCS, vol. 507, pp. 1–7. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 248–261. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Christoph Buchheim
    • 1
  • Michael Jünger
    • 1
  1. 1.Institut für InformatikUniversität zu KölnKölnGermany

Personalised recommendations