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The Weighted Arborescence Constraint

  • Vinasetan Ratheil HoundjiEmail author
  • Pierre Schaus
  • Mahouton Norbert Hounkonnou
  • Laurence Wolsey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10335)

Abstract

For a directed graph, a Minimum Weight Arborescence (MWA) rooted at a vertex r is a directed spanning tree rooted at r with the minimum total weight. We define the MinArborescence constraint to solve constrained arborescence problems (CAP) in Constraint Programming (CP). A filtering based on the LP reduced costs requires \(O(|V|^2)\) where |V| is the number of vertices. We propose a procedure to strengthen the quality of the LP reduced costs in some cases, also running in \(O(|V|^2)\). Computational results on a variant of CAP show that the additional filtering provided by the constraint reduces the size of the search tree substantially.

Keywords

Directed Graph Leaf Node Search Tree Minimum Span Tree Travel Salesman Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Beldiceanu, N., Flener, P., Lorca, X.: The tree Constraint. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 64–78. Springer, Heidelberg (2005). doi: 10.1007/11493853_7 CrossRefGoogle Scholar
  2. 2.
    Bock, F.: An algorithm to construct a minimum directed spanning tree in a directed network. Dev. Oper. Res. 1, 29–44 (1971)zbMATHGoogle Scholar
  3. 3.
    Chu, Y.J., Liu, T.H.: On the shortest arborescence of a directed graph. Sci. Sin. Ser. A 14, 1396–1400 (1965)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dooms, G., Katriel, I.: The minimum spanning tree constraint. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 152–166. Springer, Heidelberg (2006). doi: 10.1007/11889205_13 CrossRefGoogle Scholar
  5. 5.
    Dooms, G., Katriel, I.: The “not-too-heavy spanning tree” constraint. In: Hentenryck, P., Wolsey, L. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 59–70. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72397-4_5 CrossRefGoogle Scholar
  6. 6.
    Edmonds, J.: Optimum branchings. J. Res. Nat. Bur. Stand. B 71(4), 125–130 (1967)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fahle, T., Sellmann, M.: Cost based filtering for the constrained knapsack problem. Ann. Oper. Res. 115(1–4), 73–93 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fischetti, M., Toth, P.: An additive bounding procedure for asymmetric travelling salesman problem. Math. Program. 53, 173–197 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fischetti, M., Toth, P.: An efficient algorithm for min-sum arborescence problem on complete digraphs. Manage. Sci. 9(3), 1520–1536 (1993)zbMATHGoogle Scholar
  10. 10.
    Fischetti, M., Vigo, D.: A branch-and-cut algorithm for the resource-constrained minimum-weight arborescence problem. Network 29, 55–67 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Focacci, F., Lodi, A., Milano, M., Vigo, D.: Solving TSP through the integration of OR and CP techniques. Electron. Notes Discrete Math. 1, 13–25 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gabow, H.N., Galil, Z., Spencer, T.H., Tarjan, R.E.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(3), 109–122 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Hist. Comput. 7, 13–25 (1985)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Guignard, M., Rosenwein, M.B.: An application of lagrangean decomposition to the resource-constrained minimum weighted arborescence problem. Network 20, 345–359 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Houndji, V.R., Schaus, P.: Cp4cap: Constraint programming for constrained arborescence problem. https://bitbucket.org/ratheilesse/cp4cap
  16. 16.
    Kleinberg, J., Tardos, E.: Minimum-cost arborescences: a multi-phase greedy algorithm. In: Algorithm Design, Tsinghua University Press (2005)Google Scholar
  17. 17.
    Lorca, X.: Contraintes de Partitionnement de Graphe. Ph. D. thesis, Université de Nantes (2010)Google Scholar
  18. 18.
    Fages, J.-G., Lorca, X.: Revisiting the tree constraint. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 271–285. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-23786-7_22 CrossRefGoogle Scholar
  19. 19.
    Mendelson, R., Tarjan, R.E., Thorup, M., Zwick, U.: Melding priority queues. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 223–235. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-27810-8_20 CrossRefGoogle Scholar
  20. 20.
    Pesant, G., Gendreau, M., Potvin, J.-Y., Rousseau, J.-M.: An exact constraint logic programming algorithm for the traveling salesman problem with time windows. Transp. Sci. 32(1), 12–29 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Régin, J.-C.: Simpler and incremental consistency checking and arc consistency filtering algorithms for the weighted spanning tree constraint. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 233–247. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-68155-7_19 CrossRefGoogle Scholar
  22. 22.
    Régin, J.-C., Rousseau, L.-M., Rueher, M., van Hoeve, W.-J.: The weighted spanning tree constraint revisited. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 287–291. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13520-0_31 CrossRefGoogle Scholar
  23. 23.
    Tarjan, R.E.: Finding optimum branchings. Networks 7(3), 25–35 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    OscaR Team. Oscar: Scala in or (2012). https://bitbucket.org/oscarlib/oscar
  25. 25.
    Van Cauwelaert, S., Lombardi, M., Schaus, P.: Understanding the potential of propagators. In: Michel, L. (ed.) CPAIOR 2015. LNCS, vol. 9075, pp. 427–436. Springer, Cham (2015). doi: 10.1007/978-3-319-18008-3_29 Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Vinasetan Ratheil Houndji
    • 1
    • 2
    Email author
  • Pierre Schaus
    • 1
  • Mahouton Norbert Hounkonnou
    • 2
  • Laurence Wolsey
    • 1
  1. 1.Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Université d’Abomey-CalaviAbomey-CalaviBenin

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