Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Combining restarts, nogoods and bag-connected decompositions for solving CSPs

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

From a theoretical viewpoint, the (tree-)decomposition methods offer a good approach for solving Constraint Satisfaction Problems (CSPs) when their (tree)-width is small. In this case, they have often shown their practical interest. So, the literature (coming from Mathematics, OR or AI) has concentrated its efforts on the minimization of a single parameter, namely the tree-width. Nevertheless, experimental studies have shown that this parameter is not always the most relevant to consider when solving CSPs. So, in this paper, we highlight two fundamental problems related to the use of tree-decomposition and for which we offer two particularly appropriate solutions. First, we experimentally show that the decomposition algorithms of the state of the art produce clusters (a tree-decomposition is a rooted tree of clusters) having several connected components. We highlight the fact that such clusters create a real disadvantage which affects significantly the efficiency of solving methods. To avoid this problem, we consider here a new graph decomposition called Bag-Connected Tree-Decomposition, which considers only tree-decompositions such that each cluster is connected. We analyze such decompositions from an algorithmic point of view, especially in order to propose a first polynomial time algorithm to compute them. Moreover, even if we consider a very well suited decomposition, it is well known that sometimes, a bad choice for the root cluster may significantly degrade the performance of the solving. We highlight an explanation of this degradation and we propose a solution based on restart techniques. Then, we present a new version of the BTD algorithm (for Backtracking with Tree-Decomposition Jégou and Terrioux, Artificial Intelligence, 146 43–75 28) integrating restart techniques. From a theoretical viewpoint, we prove that reduced nld-nogoods can be safely recorded during the search and that their size is smaller than ones recorded by MAC+RST+NG (Lecoutre et al., JSAT, 1(3–4) 147–167 34). We also show how structural (no)goods may be exploited when the search restarts from a new root cluster. Finally, from a practical viewpoint, we show experimentally the benefits of using independently bag-connected tree-decompositions and restart techniques for solving CSPs by decomposition methods. Above all, we experimentally highlight the advantages brought by exploiting jointly these improvements in order to respond to two major problems generally encountered when solving CSPs by decomposition methods.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Explore related subjects

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

Notes

  1. We use the term “bag” rather than “cluster” because it is more compatible with the terminology of Graph Theory.

  2. We assume that E i ∩E p(i) = ∅ if E i is the root cluster.

  3. See http://www.cril.univ-artois.fr/CPAI08 for more details.

  4. For any Y⊆X, the subgraph G[Y] of G=(X,C) induced by Y is the graph (Y,C Y ) where C Y ={{x,y}∈C|x,y∈Y}.

  5. Note that we use the term of Bag-Connected Tree-Width rather than one of Connected Tree-Width exploited in [36] because the term of Connected Tree-Width has been introduced before in [15] but corresponds to a quite different concept.

  6. 6 A cycle is said geodesic if for any pair of vertices x and y belonging to the cycle, the distance between x and y in the graph is equal to the length of the shortest path between x and y in the cycle.

  7. 7 Note that this hardware configuration is very different from one used in [29, 30].

References

  1. Arnborg, S., Corneil, D., & Proskuroswki, A. (1987). Complexity of finding embeddings in a k-tree. SIAM Journal of Discrete Mathematics, 8, 277–284.

    Article  MathSciNet  MATH  Google Scholar 

  2. Berge, C. (1973). Graphs and Hypergraphs. Elsevier.

  3. Bessière, C., Meseguer, P., Freuder, E.C., & Larrosa, J. (2002). On forward checking for non-binary constraint satisfaction. Artificial Intelligence, 141, 205–224.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bessière, C., & Régin, J.C. (2001). Refining the basic constraint propagation algorithm. In Proceedings of IJCAI (pp. 309–315).

  5. Boussemart, F., Hemery, F., Lecoutre, C., & Sais, L. (2004). Boosting systematic search by weighting constraints. In Proceedings of ECAI (pp. 146–150).

  6. Cabon, C, de Givry, S., Lobjois, L., Schiex, T., & Warners, J.P. (1999). Radio link frequency assignment. Constraints, 4, 79–89.

    Article  MATH  Google Scholar 

  7. Dechter, R. (2003). Constraint processing. Morgan Kaufmann Publishers.

  8. Dechter, R., & Fattah, Y.E. (2001). Topological parameters for time-space tradeoff. Artificial Intelligence, 125, 93–118.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dechter, R., & Mateescu, R. (2004). The impact of AND/OR search spaces on constraint satisfaction and counting. In Proceedings of the 10th international conference on principles and practice of constraint programming (CP) (pp. 731–736).

  10. Dechter, R., & Mateescu, R. (2007). AND/OR search spaces for graphical models. Artificial Intelligence, 171, 73–106.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Dermaku, A., Ganzow, T., Gottlob, G., McMahan, B.J., Musliu, N., & Samer, M. (2008). Heuristic methods for hypertree decomposition. In Proceedings of MICAI (pp. 1–11).

  13. Diestel, R., & Müller, M. (2014). Connected tree-width. arXiv:1211.7353v2.

  14. Favier, A, de Givry, S., & Jégou, P. (2009). Exploiting problem structure for solution counting. In Proceedings of CP (pp. 335–343).

  15. Fraigniaud, P., & Nisse, N. (2006). Connected treewidth and connected graph searching. In Proceedings of LATIN (pp. 479–490).

  16. Freuder, E. (1982). A sufficient condition for backtrack-free search. Journal of the ACM, 29(1), 24–32.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gavril, F. (1972). Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing, 1(2), 180–187.

    Article  MathSciNet  MATH  Google Scholar 

  18. Golumbic, M. (1980). Algorithmic graph theory and perfect graphs. New York: Academic Press.

    MATH  Google Scholar 

  19. Gomes, C.P., Selman, B., Crato, N., & Kautz, H.A. (2000). Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. Journal of Automated Reasoning, 24(1/2), 67–100.

    Article  MathSciNet  MATH  Google Scholar 

  20. Gottlob, G., Leone, N., & Scarcello, F. (2000). A comparison of structural CSP decomposition methods. Artificial Intelligence, 124, 243–282.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gyssens, M., Jeavons, P., & Cohen, D. (1994). Decomposing constraint satisfaction problems using database techniques. Artificial Intelligence, 66, 57–89.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hamann, M., & Weißauer, D. (2015). Bounding connected tree-width. ArXiv:1503.01592.

  23. Harvey, W.D. (1995). Nonsystematic backtracking search. Ph.D. thesis, Stanford University.

  24. Jégou, P., Ndiaye, S.N., & Terrioux, C. (2005). Computing and exploiting tree-decompositions for solving constraint networks. In Proceedings of CP (pp. 777–781).

  25. Jégou, P., Ndiaye, S.N., & Terrioux, C. (2006). An extension of complexity bounds and dynamic heuristics for tree-decompositions of CSP. In Proceedings of CP (pp. 741–745).

  26. Jégou, P., Ndiaye, S.N., & Terrioux, C. (2007). Dynamic heuristics for backtrack search on tree-decomposition of CSPs. In Proceedings of IJCAI (pp. 112–117).

  27. Jégou, P., Ndiaye, S.N., & Terrioux, C. (2007). Dynamic management of heuristics for solving structured CSPs. In Proceedings of CP (pp. 364–378).

  28. Jégou, P., & Terrioux, C. (2003). Hybrid backtracking bounded by tree-decomposition of constraint networks. Artificial Intelligence, 146, 43–75.

    Article  MathSciNet  MATH  Google Scholar 

  29. Jėgou, P., & Terrioux, C. (2014). Combining restarts, nogoods and decompositions for solving csps. In Proceedings of ECAI (pp. 465–470).

  30. Jėgou, P., & Terrioux, C. (2014). Tree-decompositions with connected clusters for solving constraint networks. In Proceedings of CP (pp. 407–423).

  31. Karakashian, S., Woodward, R., & Choueiry, B.Y. (2013). Improving the performance of consistency algorithms by localizing and bolstering propagation in a tree decomposition. In Proceedings of AAAI (pp. 466–473).

  32. Kjaerulff, U. (1990). Triangulation of graphs - algorithms giving small total state space. Tech. rep., Judex R.R. Aalborg, Denmark.

  33. Lecoutre, C. (2009). Constraint networks - techniques and algorithms. ISTE/Wiley.

  34. Lecoutre, C., Saïs, L., Tabary, S., & Vidal, V. (2007). Recording and minimizing nogoods from restarts. JSAT, 1(3–4), 147–167.

    MATH  Google Scholar 

  35. Luby, M., Sinclair, A., & Zuckerman, D. (1993). Optimal speedup of Las Vegas algorithms. Information Processing Letters, 47(4), 173–180.

    Article  MathSciNet  MATH  Google Scholar 

  36. Müller, M. (2012). Connected tree-width. ArXiv:1211.7353.

  37. Nadel, B. (1988). Tree search and arc consistency in constraint-satisfaction algorithms. In Search in artificial intelligence (pp. 287–342). Springer-Verlag.

  38. Petke, J. (2012). On the bridge between constraint satisfaction and Boolean satisfiability. Ph.D. thesis, University of Oxford.

  39. Robertson, N., & Seymour, P. (1986). Graph minors II: algorithmic aspects of treewidth. Algorithms, 7, 309–322.

    Article  MathSciNet  MATH  Google Scholar 

  40. Rose, D.J. (1973). A graph theoretic study of the numerical solution of sparse positive denite systems of linear equations. In Read, R.C. (Ed.) Graph theory and computing (pp. 183–217). New York: Academic Press.

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

  42. Sabin, D., & Freuder, E. (1994). Contradicting conventional wisdom in constraint satisfaction. In Proceedings of ECAI (pp. 125–129).

  43. Sabin, D., & Freuder, E. (1997). Understanding and improving the MAC algorithm. In Proceedings of CP (pp. 167–181).

  44. Sanchez, M., Bouveret, S, de Givry, S., Heras, F., Jégou, P., Larrosa, J., Ndiaye, S.N., Rollon, E., Schiex, T., Terrioux, C., Verfaillie, G., & Zytnicki, M. (2008). Max-CSP competition 2008: toulbar2 solver description. In Proceedings of the 3rd CSP solver competition, CP workshop (pp. 63–70).

  45. Tarjan, R., & Yannakakis, M. (1984). Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3), 566–579.

    Article  MathSciNet  MATH  Google Scholar 

  46. Walsh, T. (1999). Search in a small world. In Proceedings of IJCAI (pp. 1172–1177).

Download references

Acknowledgments

This work was supported by the French National Research Agency under grant TUPLES (ANR-2010-BLAN-0210). The authors would like to thank Ioan Todinca for their fruitful discussion about this work.

Author information

Authors and Affiliations

  1. Aix Marseille Université, CNRS, ENSAM, Université de Toulon, LSIS UMR 7296, 13397, Marseille, France

    Philippe Jégou & Cyril Terrioux

Authors
  1. Philippe Jégou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Terrioux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyril Terrioux.

Additional information

This paper is an extension of the works published in [29, 30].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jégou, P., Terrioux, C. Combining restarts, nogoods and bag-connected decompositions for solving CSPs. Constraints 22, 191–229 (2017). https://doi.org/10.1007/s10601-016-9248-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-016-9248-8

Keywords

Search

Navigation