Skip to main content
SpringerLink
Log in
Book cover

Ordered Sets pp 113–153Cite as

Birkhäuser

Constraint Satisfaction Problems

  • Chapter
  • First Online:

  • 1645 Accesses

Abstract

When proving theorems, we have a luxury that is so fundamental, we often take it for granted: If we need to work with an object, we say “let x be \(\langle\) the object in question\(\rangle\)” and we move on with the proof. Especially in the finite setting it is obvious that, given enough patience, we should be able to find the object: Simply try out all possibilities and, if there is an object as desired, at least one of them will work. As long as we are not interested in the object itself, this approach is very efficient for developing a theory.

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   49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   99.99
Price excludes VAT (USA)
  • Durable hardcover 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

Notes

  1. 1.

    This is a possible trap when implementing algorithms or analyzing their complexity. Simple-looking steps can become quite complex and orders of magnitude of the actual complexity can be overlooked because they are hidden in sub-steps whose length does depend on the input size.

  2. 2.

    Generating every possible order-preserving self-map and checking it for a fixed point would be an exceedingly naive idea.

  3. 3.

    Again, there will be no confusion with the use of the letter \(\mathcal{C}\) here.

  4. 4.

    The ambiguity resulting from specification of constraints for sets {x i ,x j } of variables (the order of the variables matters in the specification of C ij ) is removed by demanding i < j. The alternative, which makes things unnecessarily technical, would be to specify constraints for ordered pairs of variables and demand the appropriate symmetry for the constraints C ij and C ji.

  5. 5.

    Careful with notation here. In some works, an instantiation is called consistent if (ai ,a j )∉C ij, which is consistent with a constraint being something that forbids configurations.

  6. 6.

    For our purposes we can always assume that the order of the variables x 1, …, x n is the order of the indices. We will not explicitly specify the order of the value domains.

  7. 7.

    I have seen a colleague, who was not a mathematician, use a similarly inefficient approach in his research. This is another instance that shows how mathematicians can help in collaborations.

  8. 8.

    Personally, I prefer forward checking over backtracking and I don’t think I’m the only one.

References

  1. Aho, A. V., Hopcroft, J. E., & Ullman, J. D. (1983). Data structures and algorithms. Reading: Addison-Wesley.

    MATH  Google Scholar 

  2. Bacchus, F., & Grove, A. (1995). On the forward checking algorithm. In Principles and practices in constraint programming (CP-95). Lecture notes in computer science (Vol. 976, pp. 292–309). Berlin: Springer. Available at http://www.cs.toronto.edu/~fbacchus/on-line.html

  3. Bouchitté, V., & Habib, M. (1989). The calculation of invariants for ordered sets. In I. Rival (Ed.), Algorithms and order (pp. 231–279). Dordrecht: Kluwer Academic.

    Chapter  Google Scholar 

  4. Chen, P. C. (1992). Heuristic sampling: A method for predicting the performance of tree searching programs. SIAM Journal on Computing, 21, 295–315.

    Article  MATH  Google Scholar 

  5. Clay Mathematics Institute web site. (2015). http://www.claymath.org/millennium-problems

  6. Debruyne, R., & Bessière, C. (1997). Some practicable filtering techniques for the constraint satisfaction problem. In Proceedings of the 15th International Joint Conference on Artificial Intelligence (IJCAI) (pp. 412–417).

    Google Scholar 

  7. Dechter, R. (1992). Constraint networks. In S. Shapiro (Ed.), Encyclopedia of artificial intelligence (pp. 276–284). New York: Wiley.

    Google Scholar 

  8. Dechter, R., & Pearl, J. (1989). Tree clustering for constraint networks. Artificial Intelligence, 38, 353–366.

    Article  MathSciNet  MATH  Google Scholar 

  9. Donalies, M., & Schröder, B. (2000). Performance guarantees and applications for Xia’s algorithm. Discrete Mathematics, 213, 67–86 (Proceedings of the Banach Center Minisemester on Discrete Mathematics, Week on Ordered Sets).

    Google Scholar 

  10. Freuder, E. C. (1982). A sufficient condition for backtrack-free search. Journal of the ACM, 29, 24–32.

    Article  MathSciNet  MATH  Google Scholar 

  11. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: Freeman.

    MATH  Google Scholar 

  12. Gaschnig, J. (1978). Experimental case studies of backtrack vs. (Waltz)-type vs. new algorithms for satisfying assignment problems. In Proceedings of the Second Canadian Conference on Artificial Intelligence, Toronto (pp. 268–277).

    Google Scholar 

  13. Gottlob, G. (2012). On minimal constraint networks. Artificial Intelligence, 191/192, 42–60.

    Google Scholar 

  14. Grant, K., Nowakowski, R., & Rival, I. (1995). The endomorphism spectrum of an ordered set. Order, 12, 45–55.

    Article  MathSciNet  MATH  Google Scholar 

  15. Habib, M., Morvan, M., & Rampon, J.-X. (1993). On the calculation of transitive reduction-closure of orders. Discrete Mathematics, 111, 289–303.

    Article  MathSciNet  MATH  Google Scholar 

  16. Han, C.-C., & Lee, C.-H. (1988). Comments on Mohr and Henderson’s path consistency algorithm. Artificial Intelligence, 36, 125–130.

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  18. Hogg, T., Huberman, B., & Williams, C. (1996). Phase transitions and the search problem. Artificial Intelligence, 81, 1–15.

    Article  MathSciNet  Google Scholar 

  19. Hopcroft, J., & Karp, R. (1973). A \(n^{\frac{5} {2} }\) algorithm for maximum matching in bipartite graphs. SIAM Journal on Computing, 2, 225–231.

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeavons, P., Cohen, D., & Pearson, J. (1998). Constraints and universal algebra. Annals of Mathematics and Artificial Intelligence, 24, 51–67.

    Article  MathSciNet  MATH  Google Scholar 

  21. Knuth, D. E. (1975). Estimating the efficiency of backtrack programs. Mathematics of Computation, 29, 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  22. Kondrak, G., & van Beek, P. (1997). A theoretical evaluation of selected backtracking algorithms. Artificial Intelligence, 89, 365–387.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kumar, V. (1992). Algorithms for constraint satisfaction problems – A survey. AI magazine, 13, 32–44.

    Google Scholar 

  24. Mackworth, A. K. (1992). Constraint satisfaction. In S. Shapiro (Ed.), Encyclopedia of artificial intelligence (pp. 284–293). New York: Wiley.

    Google Scholar 

  25. McGregor, J. J. (1979). Relational consistency algorithms and their application in finding subgraph and graph isomorphisms. Information Sciences, 19, 229–250.

    Article  MathSciNet  MATH  Google Scholar 

  26. Mohr, R., & Henderson, T. C. (1986). Arc and path consistency revisited. Artificial Intelligence, 28, 225–233.

    Article  Google Scholar 

  27. Nadel, B. (1989). Constraint satisfaction algorithms. Computational Intelligence, 5, 188–224.

    Article  Google Scholar 

  28. Niederle, J. (2008). Forbidden retracts for finite ordered sets of width at most four. Discrete Mathematics, 308, 1774–1784.

    Article  MathSciNet  MATH  Google Scholar 

  29. Priestley, H. A., & Ward, M. P. (1994). A multipurpose backtracking algorithm. Journal of Symbolic Computation, 18, 1–40.

    Article  MathSciNet  Google Scholar 

  30. Prosser, P. (1993). Hybrid algorithms for the constraint satisfaction problem. Computational Intelligence, 9, 268–299.

    Article  Google Scholar 

  31. Purdom, P. W. (1978). Tree size by partial backtracking. SIAM Journal on Computing, 7, 481–491.

    Article  MathSciNet  MATH  Google Scholar 

  32. Rossi, F., van Beek, P., & Walsh, T. (2006). Handbook of constraint programming. Amsterdam: Elsevier.

    MATH  Google Scholar 

  33. Sabin, D., & Freuder, E. C. (1994). Contradicting conventional wisdom in constraint satisfaction. In Proceedings of the 11th European Conference on Artificial Intelligence, Amsterdam (pp. 125–129).

    Google Scholar 

  34. Schröder, B. (1996). Fixed cliques and generalizations of dismantlability. In Y. Alavi et al. (Eds.), Combinatorics, graph theory, and algorithms. Proceedings of the Eighth Quadrennial International Conference in Graph Theory, Combinatorics, Algorithms and Applications (Vol. II, pp. 747–756).

    Google Scholar 

  35. Schröder, B. (2005). The automorphism conjecture for small sets and series parallel sets. Order, 22, 371–387.

    Article  MathSciNet  MATH  Google Scholar 

  36. Schröder, B. (2015). Homomorphic constraint satisfaction problem solver. http://www.math.usm.edu/schroeder/software.htm

    Google Scholar 

  37. Schröder, B. (2015). The fixed point property for ordered sets of interval dimension 2 (submitted to order).

    Google Scholar 

  38. Schröder, B. (2016). The use of retractions in the fixed point theory for ordered sets. In M. Alfuraidan, & Q. H. Ansari (Eds.), Fixed point theory and graph theory – Foundations and integrative approaches (pp. 365–417). Amsterdam: Elsevier. Chapter in the Proceedings of the Workshop on Fixed Point Theory and Applications, King Fahd University of Petroleum and Minerals, December 2014, Dhahran, Saudi Arabia.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Tsang, E. (1993). Foundations of constraint satisfaction. New York: Academic. Out of print. Now available through Amazon books on demand, see http://www.bracil.net/edward/FCS.html

    Google Scholar 

  41. van Beek, P., & Dechter, R. (1995). On the minimality of row-convex constraint networks. Journal of the ACM, 42, 543–561.

    Article  MathSciNet  MATH  Google Scholar 

  42. West, D. (1985). Parameters of partial orders and graphs: Packing, covering and representation. In I. Rival (Ed.), Graphs and orders (pp. 267–350). Dordrecht: Dordrecht-Reidel.

    Chapter  Google Scholar 

  43. Williams, C., & Hogg, T. (1994). Exploiting the deep structure of constraint problems. Artificial Intelligence, 70, 73–117.

    Article  MATH  Google Scholar 

  44. Xia, W. (1992). Fixed point property and formal concept analysis. Order, 9, 255–264.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, USA

    Bernd Schröder

Authors
  1. Bernd Schröder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing

About this chapter

Cite this chapter

Schröder, B. (2016). Constraint Satisfaction Problems. In: Ordered Sets. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29788-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation