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The WeightedCircuitsLmax Constraint

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Integration of Constraint Programming, Artificial Intelligence, and Operations Research (CPAIOR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10848))

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Abstract

The travelling salesman problem is a well-known problem that can be generalized to the m-travelling salesmen problem with min-max objective. In this problem, each city must be visited by exactly one salesman, among m travelling salesmen. We want to minimize the longest circuit travelled by a salesman. This paper generalizes the Circuit and WeightedCircuit constraints and presents a new constraint that encodes m cycles all starting from the same city and whose lengths are bounded by a variable \(L_{max}\). We propose two filtering algorithms, each based on a relaxation of the problem that uses the structure of the graph and the distances between each city. We show that this new constraint improves the solving time for the m travelling salesmen problem.

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References

  1. Beldiceanu, N., Carlsson, M., Rampon, J.: Global constraint catalog, 2nd edn (revision a). Technical report 03, SICS (2012)

    Google Scholar 

  2. Held, M., Karp, R.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)

    Article  MathSciNet  Google Scholar 

  3. Laporte, G., Nobert, Y.: A cutting planes algorithm for the m-salesmen problem. J. Oper. Res. Soc. 31, 1017–1023 (1980)

    Article  MathSciNet  Google Scholar 

  4. França, P.M., Gendreau, M., Laportt, G., Müller, F.M.: The m-traveling salesman problem with minmax objective. Transp. Sci. 29(3), 267–275 (1995)

    Article  Google Scholar 

  5. Necula, R., Breaban, M., Raschip, M.: Tackling the bi-criteria facet of multiple traveling salesman problem with ant colony systems. In: 2015 IEEE 27th International Conference on Tools with Artificial Intelligence (ICTAI), pp. 873–880. IEEE (2015)

    Google Scholar 

  6. Narasimha, K.V., Kivelevitch, E., Sharma, B., Kumar, M.: An ant colony optimization technique for solving min–max multi-depot vehicle routing problem. Swarm Evol. Comput. 13, 63–73 (2013)

    Article  Google Scholar 

  7. Somhom, S., Modares, A., Enkawa, T.: Competition-based neural network for the multiple travelling salesmen problem with minmax objective. Comput. Oper. Res. 26(4), 395–407 (1999)

    Article  MathSciNet  Google Scholar 

  8. Ali, A.I., Kennington, J.L.: The asymmetric m-travelling salesmen problem: a duality based branch-and-bound algorithm. Discret. Appl. Math. 13(2–3), 259–276 (1986)

    MATH  Google Scholar 

  9. Gromicho, J., Paixão, J., Bronco, I.: Exact solution of multiple traveling salesman problems. In: Akgül, M., Hamacher, H.W., Tüfekçi, S. (eds.) Combinatorial Optimization, pp. 291–292. Springer, Heidelberg (1992). https://doi.org/10.1007/978-3-642-77489-8_27

    Chapter  Google Scholar 

  10. Kara, I., Bektas, T.: Integer linear programming formulations of multiple salesman problems and its variations. Eur. J. Oper. Res. 174(3), 1449–1458 (2006)

    Article  MathSciNet  Google Scholar 

  11. Rao, M.R.: A note on the multiple traveling salesmen problem. Oper. Res. 28(3-part-i), 628–632 (1980)

    Google Scholar 

  12. Jonker, R., Volgenant, T.: An improved transformation of the symmetric multiple traveling salesman problem. Oper. Res. 36(1), 163–167 (1988)

    Article  MathSciNet  Google Scholar 

  13. Bektas, T.: The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 34(3), 209–219 (2006)

    Article  Google Scholar 

  14. Lauriere, J.L.: A language and a program for stating and solving combinatorial problems. Artif. Intell. 10(1), 29–127 (1978)

    Article  MathSciNet  Google Scholar 

  15. Caseau, Y., Laburthe, F.: Solving small TSPs with constraints. In: Proceedings of the 14th International Conference on Logic Programming (ICLP 1997), pp. 316–330 (1997)

    Google Scholar 

  16. 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). https://doi.org/10.1007/11889205_55

    Chapter  Google Scholar 

  17. Fages, J., Lorca, X.: Improving the asymmetric TSP by considering graph structure. Technical report 1206.3437, arxiv (2012)

    Google Scholar 

  18. Beldiceanu, N., Contejean, E.: Introducing global constraints in chip. Math. Comput. Modell. 20(12), 97–123 (1994)

    Article  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Focacci, F., Lodi, A., Milano, M.: Embedding relaxations in global constraints for solving TSP and TSPTW. Ann. Math. Artif. Intell. 34(4), 291–311 (2002)

    Article  MathSciNet  Google Scholar 

  21. Focacci, F., Lodi, A., Milano, M.: A hybrid exact algorithm for the TSPTW. INFORMS J. Comput. 14(4), 403–417 (2002)

    Article  MathSciNet  Google Scholar 

  22. Pesant, G., Gendreaul, M., Rousseau, J.-M.: GENIUS-CP: a generic single-vehicle routing algorithm. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 420–434. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0017457

    Chapter  Google Scholar 

  23. Tarjan, R.E.: Applications of path compression on balanced trees. J. ACM (JACM) 26(4), 690–715 (1979)

    Article  MathSciNet  Google Scholar 

  24. Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Ann. Hist. Comput. 7(1), 43–57 (1985)

    Article  MathSciNet  Google Scholar 

  25. Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7(1), 48–50 (1956)

    Article  MathSciNet  Google Scholar 

  26. Chin, F., Houck, D.: Algorithms for updating minimal spanning trees. J. Comput. Syst. Sci. 16(3), 333–344 (1978)

    Article  MathSciNet  Google Scholar 

  27. Demaine, E.D., Landau, G.M., Weimann, O.: On Cartesian trees and range minimum queries. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 341–353. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02927-1_29

    Chapter  MATH  Google Scholar 

  28. Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 246–251 (1983)

    Google Scholar 

  29. Necula, R., Breaban, M., Raschip, M.: Performance evaluation of ant colony systems for the single-depot multiple traveling salesman problem. In: Onieva, E., Santos, I., Osaba, E., Quintián, H., Corchado, E. (eds.) HAIS 2015. LNCS (LNAI), vol. 9121, pp. 257–268. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19644-2_22

    Chapter  Google Scholar 

  30. Fages, J.G., Prud’Homme, C.: Making the first solution good! In: ICTAI 2017 29th IEEE International Conference on Tools with Artificial Intelligence (2017)

    Google Scholar 

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Authors and Affiliations

  1. Université Laval, Québec City, Canada

    Kim Rioux-Paradis & Claude-Guy Quimper

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  1. Kim Rioux-Paradis
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  2. Claude-Guy Quimper
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Correspondence to Claude-Guy Quimper .

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  1. Carnegie Mellon University, Pittsburgh, Pennsylvania, USA

    Willem-Jan van Hoeve

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Rioux-Paradis, K., Quimper, CG. (2018). The WeightedCircuitsLmax Constraint. In: van Hoeve, WJ. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2018. Lecture Notes in Computer Science(), vol 10848. Springer, Cham. https://doi.org/10.1007/978-3-319-93031-2_35

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  • DOI: https://doi.org/10.1007/978-3-319-93031-2_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93030-5

  • Online ISBN: 978-3-319-93031-2

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