Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Functional and Logic Programming

FLOPS 2018: Functional and Logic Programming pp 182–197Cite as

Breaking Symmetries with Lex Implications

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10818))

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. 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

    Chapter  Google Scholar 

  2. 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)

    Google Scholar 

  3. Shlyakhter, I.: Generating effective symmetry-breaking predicates for search problems. Discrete Appl. Math. 155(12), 1539–1548 (2007)

    Article  MathSciNet  Google Scholar 

  4. 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

    Chapter  Google Scholar 

  5. 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)

    Google Scholar 

  6. 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)

    Google Scholar 

  7. 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

    Chapter  Google Scholar 

  8. 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

    Chapter  MATH  Google Scholar 

  9. 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

    Chapter  Google Scholar 

  10. Read, R.C.: Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Ann. Discrete Math. 2, 107–120 (1978)

    Article  MathSciNet  Google Scholar 

  11. Itzhakov, A., Codish, M.: Breaking symmetries in graph search with canonizing sets. Constraints 21(3), 357–374 (2016)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Google Scholar 

  13. 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)

    Google Scholar 

  14. Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Propagation algorithms for lexicographic ordering constraints. Artif. Intell. 170(10), 803–834 (2006)

    Article  MathSciNet  Google Scholar 

  15. 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

  16. 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)

    MathSciNet  MATH  Google Scholar 

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

  18. Gebser, M., Kaufmann, B., Schaub, T.: Conflict-driven answer set solving: from theory to practice. Artif. Intell. 187, 52–89 (2012)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)

    Article  MathSciNet  Google Scholar 

  21. The on-line encyclopedia of integer sequences. Published electronically (2010). http://oeis.org

  22. 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

    Chapter  MATH  Google Scholar 

  23. Radziszowski, S.P.: Small Ramsey numbers. Electron. J. Comb. (1994). Revision 14 January 2014

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Michael Codish & Avraham Itzhakov

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Thorsten Ehlers

  3. Department of Computing and Information Systems, The University of Melbourne, Melbourne, Australia

    Graeme Gange & Peter J. Stuckey

  4. Data61 CSIRO, Melbourne, Australia

    Peter J. Stuckey

Authors
  1. Michael Codish
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thorsten Ehlers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Graeme Gange
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Avraham Itzhakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peter J. Stuckey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thorsten Ehlers .

Editor information

Editors and Affiliations

  1. Roskilde University, Roskilde, Denmark

    John P. Gallagher

  2. Karlsruhe University of Applied Sciences, Karlsruhe, Germany

    Martin Sulzmann

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-90686-7_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-90685-0

  • Online ISBN: 978-3-319-90686-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation