Clique Cuts in Weighted Constraint Satisfaction

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10416)


In integer programming, cut generation is crucial for improving the tightness of the linear relaxation of the problem. This is relevant for weighted constraint satisfaction problems (WCSPs) in which we use approximate dual feasible solutions to produce lower bounds during search. Here, we investigate using one class of cuts in WCSP: clique cuts. We show that clique cuts are likely to trigger suboptimal behavior in the specialized algorithms that are used in WCSP for generating dual bounds and show how these problems can be corrected. At the same time, the additional structure present in WCSP allows us to slightly generalize these cuts. Finally, we show that cliques exist in instances from several benchmark families and that exploiting them can lead to substantial performance improvement.



This work has been partially funded by the french “Agence nationale de la Recherche”, reference ANR-16-C40-0028. We are grateful to the Bioinfo Genotoul platform Toulouse Midi-Pyrenees for providing computing resources.


  1. 1.
    Allouche, D., de Givry, S., Katsirelos, G., Schiex, T., Zytnicki, M.: Anytime hybrid best-first search with tree decomposition for weighted CSP. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 12–29. Springer, Cham (2015). doi: 10.1007/978-3-319-23219-5_2 Google Scholar
  2. 2.
    Allouche, D., Bessière, C., Boizumault, P., de Givry, S., Gutierrez, P., Lee, J.H., Leung, K.L., Loudni, S., Métivier, J.P., Schiex, T., Wu, Y.: Tractability-preserving transformations of global cost functions. Artif. Intell. 238, 166–189 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anstegui Gil, C.: Complete SAT solvers for many-valued CNF formulas. Ph.D. thesis, University of Lleida (2004)Google Scholar
  4. 4.
    Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.: Conflict graphs in solving integer programming problems. Eur. J. Oper. Res. 121(1), 40–55 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Biere, A., Le Berre, D., Lonca, E., Manthey, N.: Detecting cardinality constraints in CNF. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 285–301. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_22 Google Scholar
  6. 6.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)CrossRefMATHGoogle Scholar
  7. 7.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cooper, M., de Givry, S., Sanchez, M., Schiex, T., Zytnicki, M.: Virtual arc consistency for weighted CSP. In: Proceedings of AAAI-2008, Chicago, IL (2008)Google Scholar
  9. 9.
    Cooper, M., de Givry, S., Sanchez, M., Schiex, T., Zytnicki, M., Werner, T.: Soft arc consistency revisited. Artif. Intell. 174(7–8), 449–478 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Davies, J., Bacchus, F.: Solving MAXSAT by solving a sequence of simpler SAT instances. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 225–239. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-23786-7_19 CrossRefGoogle Scholar
  11. 11.
    Davies, J., Bacchus, F.: Postponing optimization to speed up MAXSAT solving. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 247–262. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40627-0_21 CrossRefGoogle Scholar
  12. 12.
    Dechter, R., Mateescu, R.: AND/OR search spaces for graphical models. Artif. Intell. 171(2), 73–106 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dixon, H.E.: Automating psuedo-Boolean inference within a DPLL framework. Ph.D. thesis, University of Oregon (2004)Google Scholar
  14. 14.
    de Givry, S., Prestwich, S.D., O’Sullivan, B.: Dead-end elimination for weighted CSP. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 263–272. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40627-0_22 CrossRefGoogle Scholar
  15. 15.
    de Givry, S., Schiex, T., Verfaillie, G.: Exploiting tree decomposition and soft local consistency in weighted CSP. In: Proceedings of AAAI-2006, Boston, MA (2006)Google Scholar
  16. 16.
    de Givry, S., Zytnicki, M., Heras, F., Larrosa, J.: Existential arc consistency: getting closer to full arc consistency in weighted CSPs. In: Proceedings of IJCAI-2005, Edinburgh, Scotland, pp. 84–89 (2005)Google Scholar
  17. 17.
    Globerson, A., Jaakkola, T.: Fixing max-product: convergent message passing algorithms for MAP LP-relaxations. In: Proceedings of NIPS, Vancouver, Canada (2007)Google Scholar
  18. 18.
    van Hoeve, W.J., Pesant, G., Rousseau, L.: On global warming: flow-based soft global constraints. J. Heuristics 12(4–5), 347–373 (2006)CrossRefMATHGoogle Scholar
  19. 19.
    Hurley, B., O’Sullivan, B., Allouche, D., Katsirelos, G., Schiex, T., Zytnicki, M., de Givry, S.: Multi-language evaluation of exact solvers in graphical model discrete optimization. Constraints 21(3), 413–434 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jégou, P., Terrioux, C.: Hybrid backtracking bounded by tree-decomposition of constraint networks. Artif. Intell. 146(1), 43–75 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kask, K., Dechter, R.: Branch and bound with mini-bucket heuristics. In: Proceedings of IJCAI-1999. vol. 99, pp. 426–433 (1999)Google Scholar
  22. 22.
    Khemmoudj, M.O.I., Bennaceur, H.: Clique inference process for solving Max-CSP. Eur. J. Oper. Res. 199(3), 665–673 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. IEEE Pattern Anal. Mach. Intell. 28(10), 1568–1583 (2006)CrossRefGoogle Scholar
  24. 24.
    Larrosa, J.: Boosting search with variable elimination. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 291–305. Springer, Heidelberg (2000). doi: 10.1007/3-540-45349-0_22 CrossRefGoogle Scholar
  25. 25.
    Larrosa, J., Schiex, T.: In the quest of the best form of local consistency for weighted CSP. In: Proceedings of 18th IJCAI, Acapulco, Mexico, pp. 239–244 (2003)Google Scholar
  26. 26.
    Lecoutre, C., Roussel, O., Dehani, D.E.: WCSP integration of soft neighborhood substitutability. In: Milano, M. (ed.) CP 2012. LNCS, pp. 406–421. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33558-7_31 CrossRefGoogle Scholar
  27. 27.
    Lee, J.H.M., Leung, K.L.: Consistency techniques for global cost functions in weighted constraint satisfaction. J. Artif. Intell. R. 43, 257–292 (2012)MATHGoogle Scholar
  28. 28.
    Lee, J.H., Leung, K.L.: A stronger consistency for soft global constraints in weighted constraint satisfaction. In: Proceedings of AAAI-2010, Atlanta, USA (2010)Google Scholar
  29. 29.
    Marinescu, R., Dechter, R.: AND/OR branch-and-bound for graphical models. In: Proceedings of IJCAI-2005, Edinburgh, Scotland, UK, pp. 224–229 (2005)Google Scholar
  30. 30.
    Narodytska, N., Bacchus, F.: Maximum satisfiability using core-guided MAXSAT resolution. In: Proceedings of AAAI-2014, Quebec City, Canada, pp. 2717–2723 (2014)Google Scholar
  31. 31.
    Nguyen, H., Bessiere, C., de Givry, S., Schiex, T.: Triangle-based consistencies for cost function networks. Constraints 22(2), 230–264 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Prusa, D., Werner, T.: Universality of the local marginal polytope. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp. 1738–1743 (2013)Google Scholar
  33. 33.
    Quimper, C.-G., Walsh, T.: Global grammar constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 751–755. Springer, Heidelberg (2006). doi: 10.1007/11889205_64 CrossRefGoogle Scholar
  34. 34.
    Quimper, C., Walsh, T.: Decompositions of grammar constraints. CoRR abs/0903.0470 (2009)Google Scholar
  35. 35.
    Sánchez, M., de Givry, S., Schiex, T.: Mendelian error detection in complex pedigrees using weighted constraint satisfaction techniques. Constraints 13(1), 130–154 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Schlesinger, M.I.: Syntactic analysis of two-dimensional visual signals in noisy conditions. Kibernetika 4, 113–130 (1976). (in Russian)Google Scholar
  37. 37.
    Sontag, D., Choe, D., Li, Y.: Efficiently searching for frustrated cycles in MAP inference. In: Proceedings of UAI, pp. 795–804 (2012)Google Scholar
  38. 38.
    Sontag, D., Meltzer, T., Globerson, A., Weiss, Y., Jaakkola, T.: Tightening LP relaxations for MAP using message-passing. In: Proceedings of UAI, pp. 503–510 (2008)Google Scholar
  39. 39.
    Werner, T.: A linear programming approach to max-sum problem: a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1165–1179 (2007). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MIAT, UR-875, INRACastanet TolosanFrance

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