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Static and dynamic structural symmetry breaking

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Abstract

We reconsider the idea of structural symmetry breaking for constraint satisfaction problems (CSPs). We show that the dynamic dominance checks used in symmetry breaking by dominance-detection search for CSPs with piecewise variable and value symmetries have a static counterpart: there exists a set of constraints that can be posted at the root node and that breaks all the compositions of these (unconditional) symmetries. The amount of these symmetry-breaking constraints is linear in the size of the problem, and yet they are able to remove a super-exponential number of symmetries on both values and variables. Moreover, we compare the search trees under static and dynamic structural symmetry breaking when using fixed variable and value orderings. These results are then generalised to wreath-symmetric CSPs with both variable and value symmetries. We show that there also exists a polynomial-time dominance-detection algorithm for this class of CSPs, as well as a linear-sized set of constraints that breaks these symmetries statically.

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References

  1. Cameron, P.: Permutation Groups. Number 45 in London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  2. Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of KR’96, pp. 148–159. Morgan Kaufmann, San Francisco (1996)

    Google Scholar 

  3. Fahle, T., Schamberger, S., Sellmann, M.: Symmetry breaking. In: Walsh, T. (ed.) Proceedings of CP’01. LNCS, vol. 2239, pp. 93–107. Springer, New York (2001)

    Google Scholar 

  4. Flener, P., Frisch, A.M., Hnich, B., Kızıltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetries in matrix models. In: Van Hentenryck, P. (ed.) Proceedings of CP’02. LNCS, vol. 2470, pp. 462–476. Springer, New York (2002)

    Google Scholar 

  5. Flener, P., Pearson, J., Sellmann, M., Van Hentenryck, P.: Static and dynamic structural symmetry breaking. In: Benhamou, F. (ed.) Proceedings of CP’06. LNCS, vol. 4204, pp. 695–699. Springer, New York (2006)

    Google Scholar 

  6. Flener, P., Pearson, J., Sellmann, M., Van Hentenryck, P., Ågren, M.: Dynamic structural symmetry breaking for constraint satisfaction problems. Constraints 14(4), 506–538 (2009, Union and extension of [18] and [17])

    Article  MATH  MathSciNet  Google Scholar 

  7. Focacci, F., Milano, M.: Global cut framework for removing symmetries. In: Walsh, T. (ed.) Proceedings of CP’01. LNCS, vol. 2239, pp. 77–92. Springer, New York (2001)

    Google Scholar 

  8. Gent, I.P., Smith, B.M.: Symmetry breaking during search in constraint programming. In: Proceedings of ECAI’00, pp. 599–603. IOS, Amsterdam (2000)

    Google Scholar 

  9. Heller, D., Panda, A., Sellmann, M., Yip, J.: Model restarts for structural symmetry breaking. In: Stuckey, P.J. (ed.) Proceedings of CP’08. LNCS, vol. 5202, pp. 539–544. Springer, New York (2008)

    Google Scholar 

  10. Puget, J.-F.: On the satisfiability of symmetrical constrained satisfaction problems. In: Komorowski, J., Raś, Z. (eds.) Proceedings of ISMIS’93. LNAI, vol. 689, pp. 350–361. Springer, New York (1993)

    Google Scholar 

  11. Puget, J.-F.: Constraint programming next challenge: simplicity of use. In: Wallace, M. (ed.) Proceedings of CP’04. LNCS, vol. 3258, pp. 5–8. Springer, New York (2004)

    Google Scholar 

  12. Puget, J.-F.: Automatic detection of variable and value symmetries. In: van Beek, P. (ed.) Proceedings of CP’05. LNCS, vol. 3709, pp. 475–489. Springer, New York (2005)

    Google Scholar 

  13. Puget, J.-F.: An efficient way of breaking value symmetries. In: Proceedings of AAAI’06. AAAI, Menlo Park (2006)

    Google Scholar 

  14. Puget, J.-F.: Dynamic lex constraints. In: Benhamou, F. (ed.) Proceedings of CP’06. LNCS, vol. 4204, pp. 453–467. Springer, New York (2006)

    Google Scholar 

  15. Régin, J.-C.: Generalized arc-consistency for global cardinality constraint. In: Proceedings of AAAI’96, pp. 209–215. AAAI, Menlo Park (1996)

    Google Scholar 

  16. Roney-Dougal, C.M., Gent, I.P., Kelsey, T., Linton, S.: Tractable symmetry breaking using restricted search trees. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of ECAI’04, pp. 211–215. IOS, Amsterdam (2004)

    Google Scholar 

  17. Sellmann, M., Van Hentenryck, P.: Structural symmetry breaking. In: Proceedings of IJCAI’05 (2005)

  18. Van Hentenryck, P., Flener, P., Pearson, J., Ågren, M.: Tractable symmetry breaking for CSPs with interchangeable values. In: Proceedings of IJCAI’03, pp. 277–282. Morgan Kaufmann, San Francisco (2003)

    Google Scholar 

  19. Van Hentenryck, P., Flener, P., Pearson, J., Ågren, M.: Compositional derivation of symmetries for constraint satisfaction. In: Zucker, J.-D., Saitta, L. (eds.) Proceedings of SARA’05. LNCS, vol. 3607, pp. 234–247. Springer, New York (2005)

    Google Scholar 

  20. Walsh, T.: Breaking value symmetry. In: Bessière, Ch. (ed.) Proceedings of CP’07. LNCS, vol. 4741, pp. 880–887. Springer, New York (2007)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, Tuzla, 34956, İstanbul, Turkey

    Pierre Flener

  2. Department of Information Technology, Uppsala University, Box 337, 751 05, Uppsala, Sweden

    Pierre Flener & Justin Pearson

  3. Department of Computer Science, Brown University, Box 1910, Providence, RI, 02912, USA

    Meinolf Sellmann

Authors
  1. Pierre Flener
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  2. Justin Pearson
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  3. Meinolf Sellmann
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Correspondence to Pierre Flener.

Additional information

Most of the work by Pierre Flener was done while on leave of absence in 2006/07 as a Visiting Faculty Member and Erasmus Exchange Teacher at Sabancı University.

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Flener, P., Pearson, J. & Sellmann, M. Static and dynamic structural symmetry breaking. Ann Math Artif Intell 57, 37–57 (2009). https://doi.org/10.1007/s10472-009-9172-3

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  • DOI: https://doi.org/10.1007/s10472-009-9172-3

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