Abstract
Breaking symmetries is crucial when solving hard combinatorial problems. A common way to eliminate symmetries in CP/SAT is to add symmetry breaking constraints. Ideally, symmetry breaking constraints should be complete and compact. The aim of this paper is to find compact and complete symmetry breaks applicable when solving hard combinatorial problems using CP/SAT approach. In particular: graph search problems and matrix model problems where symmetry breaks are often specified in terms of lex constraints. We show that sets of lex constraints can be expressed with only a small portion of their inner lex implications which are a particular form of Horn clauses. We exploit this fact and compute a compact encoding of the row-wise LexLeader and state of the art partial symmetry breaking constraints. We illustrate the approach for graph search problems and matrix model problems.
Supported by the Israel Science Foundation, grant 625/17 and the German Federal Ministry of Education and Research, combined project 01IH15006A.
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References
Puget, J.-F.: On the satisfiability of symmetrical constrained satisfaction problems. In: Komorowski, J., Raś, Z.W. (eds.) ISMIS 1993. LNCS, vol. 689, pp. 350–361. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56804-2_33
Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A.: Symmetry-breaking predicates for search problems. In: Aiello, L.C., Doyle, J., Shapiro, S.C. (eds.): Proceedings of the Fifth International Conference on Principles of Knowledge Representation and Reasoning (KR 1996), Cambridge, Massachusetts, USA, 5–8 November 1996, pp. 148–159. Morgan Kaufmann (1996)
Shlyakhter, I.: Generating effective symmetry-breaking predicates for search problems. Discrete Appl. Math. 155(12), 1539–1548 (2007)
Walsh, T.: General symmetry breaking constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 650–664. Springer, Heidelberg (2006). https://doi.org/10.1007/11889205_46
Walsh, T.: Symmetry breaking constraints: recent results. In: Hoffmann, J., Selman, B. (eds.) Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, Toronto, Ontario, Canada, 22–26 July 2012. AAAI Press (2012)
Codish, M., Miller, A., Prosser, P., Stuckey, P.J.: Breaking symmetries in graph representation. In: Rossi, F. (ed.) Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI 2013, Beijing, China, 3–9 August 2013, pp. 510–516. IJCAI/AAAI (2013)
Flener, P., Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetries in matrix models. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 462–477. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-46135-3_31
Frisch, A.M., Jefferson, C., Miguel, I.: Constraints for breaking more row and column symmetries. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 318–332. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45193-8_22
Grayland, A., Miguel, I., Roney-Dougal, C.M.: Snake lex: an alternative to double lex. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 391–399. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04244-7_32
Read, R.C.: Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Ann. Discrete Math. 2, 107–120 (1978)
Itzhakov, A., Codish, M.: Breaking symmetries in graph search with canonizing sets. Constraints 21(3), 357–374 (2016)
Heule, M.J.H.: The quest for perfect and compact symmetry breaking for graph problems. In: Davenport, J.H., Negru, V., Ida, T., Jebelean, T., Petcu, D., Watt, S.M., Zaharie, D. (eds.) 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2016, Timisoara, Romania, 24–27 September 2016, pp. 149–156. IEEE Computer Society (2016)
Frisch, A.M., Harvey, W.: Constraints for breaking all row and column symmetries in a three-by-two matrix. In: Proceedings of SymCon 2003 (2003)
Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Propagation algorithms for lexicographic ordering constraints. Artif. Intell. 170(10), 803–834 (2006)
Smith, B.: Symmetry breaking constraints in constraint programming (2010). Slides published online. http://ta.twi.tudelft.nl/wst/users/achill/MFOSymOpt2010/MFOSymOpt2010/Oberwolfach_Mini-Workshop_files/BarbaraMfoSlides.ppt
Metodi, A., Codish, M., Stuckey, P.J.: Boolean equi-propagation for concise and efficient SAT encodings of combinatorial problems. J. Artif. Intell. Res. (JAIR) 46, 303–341 (2013)
Audemard, G., Simon, L.: Glucose 4.0 SAT solver. http://www.labri.fr/perso/lsimon/glucose/
Gebser, M., Kaufmann, B., Schaub, T.: Conflict-driven answer set solving: from theory to practice. Artif. Intell. 187, 52–89 (2012)
Cameron, R.D., Colbourn, C.J., Read, R.C., Wormald, N.C.: Cataloguing the graphs on 10 vertices. J. Graph Theor. 9(4), 551–562 (1985)
Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)
The on-line encyclopedia of integer sequences. Published electronically (2010). http://oeis.org
Fourdrinoy, O., Grégoire, É., Mazure, B., Saïs, L.: Eliminating redundant clauses in SAT instances. In: Van Hentenryck, P., Wolsey, L. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 71–83. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72397-4_6
Radziszowski, S.P.: Small Ramsey numbers. Electron. J. Comb. (1994). Revision 14 January 2014
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Codish, M., Ehlers, T., Gange, G., Itzhakov, A., Stuckey, P.J. (2018). Breaking Symmetries with Lex Implications. In: Gallagher, J., Sulzmann, M. (eds) Functional and Logic Programming. FLOPS 2018. Lecture Notes in Computer Science(), vol 10818. Springer, Cham. https://doi.org/10.1007/978-3-319-90686-7_12
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