A Circuit Constraint for Multiple Tours Problems

  • Philippe VismaraEmail author
  • Nicolas Briot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)


Routing problems appear in many practical applications. In the context of Constraint Programming, circuit constraints have been successfully developed to handle problems like the well-known Traveling Salesman Problem or the Vehicle Routing Problem. These kind of constraints are linked to the search for a Hamiltonian circuit in a graph. In this paper we consider a more general multiple tour problem that consists in covering a part of the graph with a set of minimal cost circuits. We define a new global constraint WeightedSubCircuits that generalizes the WeightedCircuit constraint by releasing the need to obtain a Hamiltonian circuit. It enforces multiple disjoint circuits of bounded total cost to partially cover a weighted graph, the subsets of vertices to be covered being induced by external constraints. We show that enforcing Bounds Consistency for WeightedSubCircuits is NP-hard. We propose an incomplete but polynomial filtering method based on the search for a lower bound of a weighted Steiner circuit.


  1. 1.
    Toth, P., Vigo, D.: Vehicle routing: problems, methods, and applications. In: SIAM (2014)Google Scholar
  2. 2.
    Laurière, J.: A language and a program for stating and solving combinatorial problems. Artif. Intell. 10(1), 29–127 (1978)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beldiceanu, N., Contejean, E.: Introducing global constraints in chip. Math. Comput. Model. 20(12), 97–123 (1994)CrossRefGoogle Scholar
  4. 4.
    Régin, J.C.: A filtering algorithm for constraints of difference in CSPs. In: Proceedings of the National Conference on Artificial Intelligence, AAAI-94, pp. 362–367. Seattle (1994)Google Scholar
  5. 5.
    Kaya, L.G., Hooker, J.N.: A filter for the circuit constraint. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 706–710. Springer, Heidelberg (2006). Scholar
  6. 6.
    Francis, K.G., Stuckey, P.J.: Explaining circuit propagation. Constraints 19(1), 1–29 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caseau, Y., Laburthe, F.: Solving small TSPs with constraints. In: Proceedings of the 14th International Conference on Logic Programming, pp. 316–330. MIT Press (1997)Google Scholar
  8. 8.
    Pesant, G., Gendreau, M., Potvin, J., Rousseau, J.: An exact constraint logic programming algorithm for the traveling salesman problem with time windows. Transp. Sci. 32(1), 12–29 (1998)CrossRefGoogle Scholar
  9. 9.
    Dooms, G.: The CP (Graph) computation domain in constraint programming. Ph.D. thesis, Université catholique de Louvain (2006)Google Scholar
  10. 10.
    Prosser, P., Unsworth, C.: A connectivity constraint using bridges. In: ECAI 2006: 17th European Conference on Artificial Intelligence, Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 707–708. IOS Press (2006)Google Scholar
  11. 11.
    Focacci, F., Lodi, A., Milano, M.: A hybrid exact algorithm for the TSPTW. INFORMS J. Comput. 14(4), 403–417 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Benchimol, P., Hoeve, W.J.V., Régin, J.C., Rousseau, L.M., Rueher, M.: Improved filtering for weighted circuit constraints. Constraints 17(3), 205–233 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ducomman, S., Cambazard, H., Penz, B.: Alternative filtering for the weighted circuit constraint: comparing lower bounds for the TSP and solving TSPTW. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 3390–3396 (2016)Google Scholar
  14. 14.
    Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G.: MiniZinc: towards a standard CP modelling language. In: Bessiere, C. (ed.) Principles and Practice of Constraint Programming - CP 2007, pp. 529–543. Springer, Berlin Heidelberg (2007). Scholar
  15. 15.
    Hwang, F.K., Richards, D.S., Winter, P.: The Steiner tree problem. In: Annals of Discrete Mathematics, vol. 53. Elsevier (1992)Google Scholar
  16. 16.
    Di Gaspero, L., Rendl, A., Urli, T.: Constraint-based approaches for balancing bike sharing systems. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 758–773. Springer, Heidelberg (2013). Scholar
  17. 17.
    Di Gaspero, L., Rendl, A., Urli, T.: Balancing bike sharing systems with constraint programming. Constraints 21(2), 318–348 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Prud’homme, C., Fages, J.G., Lorca, X.: Choco Documentation. TASC, INRIA Rennes, LINA CNRS UMR 6241, COSLING S.A.S. (2016).

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.LIRMMUniv Montpellier, CNRSMontpellierFrance
  2. 2.MISTEA, Montpellier SupAgro, INRAUniv MontpellierMontpellierFrance

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