Advertisement

Integration of Structural Constraints into TSP Models

  • Nicolas IsoartEmail author
  • Jean-Charles Régin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11802)

Abstract

Several models based on constraint programming have been proposed to solve the traveling salesman problem (TSP). The most efficient ones, such as the weighted circuit constraint (WCC), mainly rely on the Lagrangian relaxation of the TSP, based on the search for spanning tree or more precisely “1-tree”. The weakness of these approaches is that they do not include enough structural constraints and are based almost exclusively on edge costs. The purpose of this paper is to correct this drawback by introducing the Hamiltonian cycle constraint associated with propagators. We propose some properties preventing the existence of a Hamiltonian cycle in a graph or, conversely, properties requiring that certain edges be in the TSP solution set. Notably, we design a propagator based on the research of k-cutsets. The combination of this constraint with the WCC constraint allows us to obtain, for the resolution of the TSP, gains of an order of magnitude for the number of backtracks as well as a strong reduction of the computation time.

Keywords

Global constraint TSP Propagator 

Notes

Acknowledgements

We would like to thank Pr. Tsin for sending us his 2-cutset search algorithm implementation.

References

  1. 1.
    Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  2. 2.
    Benchimol, P., Régin, J.-C., Rousseau, L.-M., Rueher, M., van Hoeve, W.-J.: Improving the held and karp approach with constraint programming. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 40–44. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13520-0_6CrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, N., Coudert, D.: Le défi des 1001 graphes. Interstices, December 2017. https://hal.inria.fr/hal-01662565
  4. 4.
    Ducomman, S., Cambazard, H., Penz, B.: Alternative filtering for the weighted circuit constraint: comparing lower bounds for the TSP and solving TSPTW. In: AAAI (2016)Google Scholar
  5. 5.
    Fages, J.G., Lorca, X., Rousseau, L.M.: The salesman and the tree: the importance of search in CP. Constraints 21(2), 145–162 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Haythorpe, M.: FHCP challenge set: the first set of structurally difficult instances of the Hamiltonian cycle problem (2019)Google Scholar
  7. 7.
    Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees: Part ii. Math. Program. 1(1), 6–25 (1971)CrossRefGoogle Scholar
  9. 9.
    Reinelt, G.: TSPLIB–A traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991)CrossRefGoogle Scholar
  10. 10.
    Sellmann, M.: Theoretical foundations of CP-based lagrangian relaxation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 634–647. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30201-8_46CrossRefzbMATHGoogle Scholar
  11. 11.
    Tarjan, R.E.: A note on finding the bridges of a graph. Inf. Process. Lett. 2, 160–161 (1974)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tarjan, R.E.: Data structures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Mathematics (1983)Google Scholar
  13. 13.
    Tsin, Y.H.: Yet another optimal algorithm for 3-edge-connectivity. J. Discrete Algorithms 7(1), 130–146 (2009). Selected papers from the 1st International Workshop on Similarity Search and Applications (SISAP)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yeh, L.P., Wang, B.F., Su, H.H.: Efficient algorithms for the problems of enumerating cuts by non-decreasing weights. Algorithmica 56(3), 297–312 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, I3SSophia AntipolisFrance

Personalised recommendations