Skip to main content

Advertisement

Springer Nature Link
Log in

On the hardness of solving edge matching puzzles as SAT or CSP problems

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

Edge matching puzzles have been amongst us for a long time now and traditionally they have been considered, both, a children’s game and an interesting mathematical divertimento. Their main characteristics have already been studied, and their worst-case complexity has been properly classified as a NP-complete problem. It is in recent times, specially after being used as the problem behind a money-prized contest, with a prize of 2US$ million for the first solver, that edge matching puzzles have attracted mainstream attention from wider audiences, including, of course, computer science people working on solving hard problems. We consider these competitions as an interesting opportunity to showcase SAT/CSP solving techniques when confronted to a real world problem to a broad audience, a part of the intrinsic, i.e. monetary, interest of such a contest. This article studies the NP-complete problem known as edge matching puzzle using SAT and CSP approaches for solving it. We will focus on providing, first and foremost, a theoretical framework, including a generalized definition of the problem. We will design and show algorithms for easy and fast problem instances generation, generators with easily tunable hardness. Afterwards we will provide with SAT and CSP models for the problems and we will study problem complexity, both typical case and worst-case complexity. We will also provide some specially crafted heuristics that result in a boost in solving time and study which is the effect of such heuristics.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Achlioptas, D., Gomes, C., Kautz, H., Selman, B. (2000). Generating satisfiable problem instances. In Proceedings of the AAAI 2000 (pp. 256–261). AAAI Press/The MIT Press.

  2. Achlioptas, D., Jia, H., Moore, C. (2005). Hiding truth assignments: two are better than one. Journal of Artifical Intelligence Research, 24, 623–639.

    MathSciNet  MATH  Google Scholar 

  3. Achlioptas, D., Naor, A., Peres, Y. (2005). Rigorous location of phase transitions in hard optimization problems. Nature, 435(7043), 759.

    Article  Google Scholar 

  4. Achlioptas, D., & Peres, Y. (2004). The threshold for random k-sat is \(2^k\log{2}-\mathcal{O}(k)\). Journal of the American Mathematical Society, 17(4), 947–973.

    Article  MathSciNet  MATH  Google Scholar 

  5. Alon, N., & Spencer, J.H. (2000). The probabilistic method (2nd ed.). Discrete Mathematics and Optimization. Wiley Inter-Science.

  6. Ansótegui, C., Béjar, R., Fernández, C., Gomes, C., Mateu, C. (2011). The impact of balance in a highly structured problem domain. In Proceedings of the AAAI 2006 (pp. 438–443). AAAI Press/The MIT Press.

  7. Ansótegui, C., Béjar, R., Fernández, C., Gomes, C., Mateu, C. (2011). Generating highly balanced sudoku problems as hard problems. Journal of Heuristics, 17(5), 589–614. doi:10.1007/s10732-010-9146-y.

    Article  Google Scholar 

  8. Ansótegui, C., Béjar, R., Fernàndez, C., Mateu, C. (2008). Edge matching puzzles as hard SAT/CSP benchmarks. In CP ’08: proceedings of the 14th international conference on principles and practice of constraint programming. Lecture notes in computer science (Vol. 5202, pp. 560–565). Sydney, Australia: Springer.

  9. Ansótegui, C., Béjar, R., Fernández, C., Mateu, C. (2008). How hard is a commercial puzzle: the Eternity II challenge. Frontiers in Artificial Intelligence and Applications - Artificial Intelligence Research and Development, 184, 99–108.

    Google Scholar 

  10. Ansótegui, C., del Val, A., Dotú, I., Fernández, C., Manyà, F. (2004). Modelling choices in quasigroup completion: SAT vs CSP. In Proceedings of the AAAI 2004. AAAI Press/The MIT Press.

  11. Ansótegui, C., & Manyà, F. (2005). Mapping many-valued CNF formulas to Boolean CNF formulas. In Proceedings of the international symposia on multiple-valued logic (pp. 290–295).

  12. Atserias, A., Bulatov, A.A., Dalmau, V. (2007). On the power of k-consistency. In 34th International Colloquium Automata, Languages and Programming, ICALP 2007. of Lecture notes in computer science (Vol. 4596, pp. 279–290). Springer.

  13. Béjar, R., Fernàndez, C., Mateu, C., Pascual, N. (2009). Bounding the phase transition on edge matching puzzles. IEEE International Symposium on Multiple-Valued Logic (pp. 80–85).

  14. Benoist, T., & Bourreau, E. (2008). Fast global filtering for eternity II. Constraint Programming Letters, 3, 36–49.

    Google Scholar 

  15. Bessière, C., & Régin, J.-C. (1996). MAC and combined heuristics: two reasons to forsake FC (and CBJ?) on hard problems. In Principles and practice of constraint programming (pp. 61–75).

  16. Cohen, D.A., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M. (2006). Symmetry definitions for constraint satisfaction problems. Constraints, 11(2–3), 115–137.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dechter, R. (2003). Constraint processing. Morgan Kaufmann.

  18. Demaine, E.D., & Demaine, M.L. (2007). Jigsaw puzzles, edge matching, and polyomino packing: connections and complexity. Graphs and Combinatorics, 23(s1), 195.

    Article  MathSciNet  MATH  Google Scholar 

  19. Eén, N., & Biere, A. (2005). Effective preprocessing in SAT through variable and clause elimination. In Proceedings of SAT 2005 (pp. 61–75).

  20. Eén, N., & Sörensson, N. (2003). An extensible SAT-solver. In Proceedings of SAT 2003. Lecture notes in computer science (Vol. 2919, pp. 502–518). Springer.

  21. Eén, N., & Sörensson, N. (2006). Translating pseudo-Boolean constraints into SAT. Journal of Satisfiability, 2, 1–26.

    MATH  Google Scholar 

  22. Gent, I.P., Jefferson, C., Miguel, I. (2006). MINION: a fast, scalable, constraint solver. In Proceedings of the 2006 conference on ECAI 2006: 17th European conference on artificial intelligence, 29 Aug–1 Sept 2006, Riva del Garda, Italy (pp. 98–102). Amsterdam, The Netherlands, The Netherlands: IOS Press.

    Google Scholar 

  23. Gent, I.P., Jefferson, C., Miguel, I. (2006). Watched literals for constraint propagation in Minion. In Principles and practice of constraint programming (pp. 182–197).

  24. Haanpää, H., Järvisalo, M., Kaski, P., Niemelä, I. (2006). Hard satisfiable clause sets for benchmarking equivalence reasoning techniques. Journal on Satisfiability, Boolean Modeling and Computation, 2(1–4), 27–46.

    MATH  Google Scholar 

  25. Haralick, R.M., & Elliott, G.L. (1980). Increasing tree search efficiency for constraint satisfaction problems. AI Journal, 14, 263–313.

    Google Scholar 

  26. Järvisalo, M. (2006). Further investigations into regular XORSAT. In Proceedings of the AAAI 2006. AAAI Press/The MIT Press.

  27. Li, C.M., & Anbulagan, A. (1997). Heuristics based on unit propagation for satisfiability problems. In Proceedings of the International Joint Conference on Artif icial Intelligence, IJCAI 97 (pp. 366–371). Morgan Kaufmann.

  28. Monasson, R., Zecchinna, R., Kirkpatrick, S., Selman, B., Troyansky, L. (1999). Determining computational complexity from characteristic phase transitions. Nature, 400, 133–137.

    Article  MathSciNet  Google Scholar 

  29. Prosser, P. (1996). An empirical study of phase transitions in binary constraint satisfaction problems. AI Journal, 81, 81–109.

    MathSciNet  Google Scholar 

  30. Régin, J.-C. (1994). A filtering algorithm for constraints of difference in CSPs. In Proceedings of the AAAI 1994 (pp. 362–367). AAAI Press/The MIT Press.

  31. Régin, J.-C. (1999). The symmetric alldiff constraint. In Proceedings of the sixteenth International Joint Conference on Artificial Intelligence, IJCAI 99 (pp. 420–425). Morgan Kaufmann.

  32. Schaus, P., & Deville, Y. (2008). Hybridization of CP and VLNS for Eternity II. In Quatrime Journes Francophones de Programmation par Contraintes (JFPC’08).

  33. Smith, B., & Dyer, M. (1996). Locating the phase transition in binary constraint satisfaction problems. Artificial Intelligence, 81, 155–181.

    Article  MathSciNet  Google Scholar 

  34. Takenaga, Y., & Walsh, T. (2006). Tetravex is NP-complete. Information Processing Letters, 99(5), 171–174.

    Article  MathSciNet  MATH  Google Scholar 

  35. Williams, R., Gomes, C.P., Selman, B. (2003). Backdoors to typical case complexity. In IJCAI (pp. 1173–1178).

  36. Xu, K., & Li, W. (2000). Exact phase transition in random constraint satisfaction problems. Journal of Artificial Intelligence Research, 12, 93–103.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universitat de Lleida, Lleida, Spain

    Carlos Ansótegui, Ramón Béjar, Cèsar Fernández & Carles Mateu

Authors
  1. Carlos Ansótegui
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Ramón Béjar
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Cèsar Fernández
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Carles Mateu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Carles Mateu.

Additional information

This is an extended version that completes previous work presented in [8, 9, 13].

Research partially supported by projects TIN2009-14704-C03-01 and TIN2010-20967-C04-03 funded by the Ministerio de Educación y Ciencia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ansótegui, C., Béjar, R., Fernández, C. et al. On the hardness of solving edge matching puzzles as SAT or CSP problems. Constraints 18, 7–37 (2013). https://doi.org/10.1007/s10601-012-9128-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-012-9128-9

Keywords