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Complete symmetry breaking constraints for the class of uniquely Hamiltonian graphs

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Abstract

This paper introduces, for the first time, a complete symmetry breaking constraint of polynomial size for a significant class of graphs: the class of uniquely Hamiltonian graphs. We introduce a canonical form for uniquely Hamiltonian graphs and prove that testing whether a given uniquely Hamiltonian graph is canonical can be performed efficiently. Based on this canonicity test, we construct a complete symmetry breaking constraint of polynomial size which is satisfied only by uniquely Hamiltonian graphs which are canonical. We apply the proposed symmetry breaking constraint to show new results regarding the class of uniquely Hamiltonian graphs. We also show that the proposed approach applies almost directly for the class of graphs which contain any cycle of known length where it shown to result in a partial symmetry breaking constraint. Given that it is unknown if there exist complete symmetry breaking constraints for graphs of polynomial size, this paper makes a first step in the direction of identifying specific classes of graphs for which such constraints do exist.

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References

  1. Audemard, G., Simon, L.: Glucose 4.0 SAT Solver. http://www.labri.fr/perso/lsimon/glucose/

  2. Bellman, R. (1962). Dynamic programming treatment of the travelling salesman problem. Journal of the ACM, 9(1), 61–63.

    Article  MathSciNet  Google Scholar 

  3. Bondy, J., & Jackson, B. (1998). Vertices of small degree in uniquely Hamiltonian graphs. Journal of Combinatorial Theory, Series B, 74(2), 265–275.

    Article  MathSciNet  Google Scholar 

  4. Cameron, R., Colbourn, C., Read, R., & Wormald, N. C. (1985). Cataloguing the graphs on 10 vertices. Journal of Graph Theory, 9(4), 551–562.

    Article  MathSciNet  Google Scholar 

  5. Codish, M., Ehlers, T., Gange, G., Itzhakov, A., & Stuckey, P. J. (2018). Breaking symmetries with lex implications. Lecture Notes in Computer ScienceIn J. P. Gallagher & M. Sulzmann (Eds.), Functional and Logic Programming - 14th International Symposium, FLOPS 2018, Nagoya, Japan, May 9–11, 2018, Proceedings (Vol. 10818, pp. 182–197). Springer.

    Chapter  Google Scholar 

  6. Codish, M., Frank, M., Itzhakov, A., & Miller, A. (2016). Computing the Ramsey number r(4, 3, 3) using abstraction and symmetry breaking. Constraints An International Journal, 21(3), 375–393.

    Article  MathSciNet  Google Scholar 

  7. Codish, M., Gange, G., Itzhakov, A., & Stuckey, P. J. (2016). Breaking symmetries in graphs: The nauty way. Lecture Notes in Computer ScienceIn M. Rueher (Ed.), Principles and Practice of Constraint Programming - 22nd International Conference, CP 2016, Toulouse, France, September 5–9, 2016, Proceedings (Vol. 9892, pp. 157–172). Springer.

    Google Scholar 

  8. Codish, M., Miller, A., Prosser, P., Stuckey, P.J. (2013). Breaking symmetries in graph representation. In F. Rossi (Ed.) IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, August 3-9, 2013 (pp. 510–516). Beijing, China, IJCAI/AAAI. http://ijcai.org/proceedings/2013

  9. Codish, M., Miller, A., Prosser, P., & Stuckey, P. J. (2019). Constraints for symmetry breaking in graph representation. Constraints, 24(1), 1–24.

    Article  MathSciNet  Google Scholar 

  10. Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A. (1996). Symmetry-breaking predicates for search problems. In L. C. Aiello, J. Doyle, S. C. Shapiro (Eds.) Proceedings of the Fifth International Conference on Principles of Knowledge Representation and Reasoning (KR’96), November 5-8, 1996.(pp. 148–159). Morgan Kaufmann, Cambridge, Massachusetts, USA

  11. Fleischner, H. (2014). Uniquely Hamiltonian graphs of minimum degree 4. Journal of Graph Theory, 75(2), 167–177.

    Article  MathSciNet  Google Scholar 

  12. Frisch, A.M., Harvey, W. (2003). Constraints for breaking all row and column symmetries in a three-by-two matrix. In Proceedings of SymCon‘03

  13. Garnick, D. K., Kwong, Y. H. H., & Lazebnik, F. (1993). Extremal graphs without three-cycles or four-cycles. Journal of Graph Theory, 17(5), 633–645.

    Article  MathSciNet  Google Scholar 

  14. Goedgebeur, J., Meersman, B., & Zamfirescu, C. T. (2019). Graphs with few Hamiltonian cycles. Mathematics of Computation, 89(322), 965–991.

    Article  MathSciNet  Google Scholar 

  15. Held, M., & Karp, R. M. (1965). The construction of discrete dynamic programming algorithms. IBM Systems Journal, 4(2), 136–147.

    Article  Google Scholar 

  16. Heule, M.J.H. (2019). Optimal symmetry breaking for graph problems. Mathematics in Computer Science

  17. Itzhakov, A., Codish, M. (2016). Breaking symmetries in graph search with canonizing sets. Constraints pp. 1–18

  18. Itzhakov, A., & Codish, M. (2020). Incremental symmetry breaking constraints for graph search problems. Proceedings of the AAAI Conference on Artificial Intelligence, 34(02), 1536–1543.

    Article  Google Scholar 

  19. McGuire, G., Tugemann, B., Civario, G. (2012). There is no 16-clue Sudoku: Solving the Sudoku minimum number of clues problem. CoRR arXiv: abs/1201.0749

  20. McKay, B. D. (1981). Practical Graph Isomorphism. Congressus Numerantium, 30, 45–87.

    MathSciNet  MATH  Google Scholar 

  21. McKay, B. D. (1998). Isomorph-free exhaustive generation. Journal of Algorithms, 26(2), 306–324.

    Article  MathSciNet  Google Scholar 

  22. McKay, B. D., & Piperno, A. (2014). Practical graph isomorphism, II. Journal of Symbolic Computation, 60, 94–112.

    Article  MathSciNet  Google Scholar 

  23. Metodi, A., & Codish, M. (2012). Compiling finite domain constraints to SAT with BEE. Theory and Practice of Logic Programming, 12(4–5), 465–483.

    Article  Google Scholar 

  24. Metodi, A., Codish, M., & Stuckey, P. J. (2013). Boolean equi-propagation for concise and efficient SAT encodings of combinatorial problems. Journal of Artificial Intelligence Research, 46, 303–341.

    Article  MathSciNet  Google Scholar 

  25. The on-line encyclopedia of integer sequences. published electronically at http://oeis.org (2010)

  26. Prud’homme, C., Fages, J.G., Lorca, X. (2016). Choco Solver Documentation. TASC, INRIA Rennes, LINA CNRS UMR 6241, COSLING S.A.S. http://www.choco-solver.org

  27. Royle, G., What is the smallest uniquely Hamiltonian graph with minimum degree at least 3? MathOverflow. https://mathoverflow.net/q/255784 (version: 2020-10-17)

  28. Seamone, B. (2015). On uniquely Hamiltonian claw-free and triangle-free graphs. Discussiones Mathematicae Graph Theory, 35(2), 207–214.

    Article  MathSciNet  Google Scholar 

  29. Sheehan, J. (1977). Graphs with exactly one Hamiltonian circuit. Journal of Graph Theory, 1(1), 37–43.

    Article  MathSciNet  Google Scholar 

  30. Shlyakhter, I. (2007). Generating effective symmetry-breaking predicates for search problems. Discrete Applied Mathematics, 155(12), 1539–1548.

    Article  MathSciNet  Google Scholar 

  31. Walsh, T. (2006). General symmetry breaking constraints. In Principles and Practice of Constraint Programming - CP 2006, 12th International Conference, CP 2006, Nantes, France, September 25-29, 2006, Proceedings(pp. 650–664)

  32. Walsh, T.: Symmetry breaking constraints: Recent results. Proceedings of the AAAI Conference on Artificial Intelligence 26(1) (2012)

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Funding

This work was supported by the Israel Science Foundation, grant 625/17.

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Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Avraham Itzhakov & Michael Codish

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  1. Avraham Itzhakov
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  2. Michael Codish
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Correspondence to Avraham Itzhakov.

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The authors have no relevant financial or non-financial interests to disclose.

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Itzhakov, A., Codish, M. Complete symmetry breaking constraints for the class of uniquely Hamiltonian graphs. Constraints 27, 8–28 (2022). https://doi.org/10.1007/s10601-021-09323-8

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