Abstract
The Traveling Salesman Problem, at the heart of many routing applications, has a few well-known relaxations that have been very effective to compute lower bounds on the objective function or even to perform cost-based domain filtering in constraint programming models. We investigate other ways of using such relaxations based on computing the frequency of edges in near-optimal solutions to a relaxation. We report early empirical results on symmetric instances from tsplib.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Concorde TSP Solver. https://en.wikipedia.org/wiki/Concorde_TSP_Solver
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). https://doi.org/10.1007/11493853_7
Benchimol, P., van Hoeve, W.J., Régin, J.-C., Rousseau, L.-M., Rueher, M.: Improved filtering for weighted circuit constraints. Constraints 17(3), 205–233 (2012)
Brockbank, S., Pesant, G., Rousseau, L.-M.: Counting spanning trees to guide search in constrained spanning tree problems. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 175–183. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40627-0_16
Broder, A.Z., Mayr, E.W.: Counting minimum weight spanning trees. J. Algorithms 24(1), 171–176 (1997)
Caseau, Y., Laburthe, F.: Solving various weighted matching problems with constraints. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 17–31. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0017427
Christofides, N., Mingozzi, A., Toth, P.: State space relaxation procedures for the computation of bounds to routing problems. Networks 11, 145–164 (1981)
de Uña, D., Gange, G., Schachte, P., Stuckey, P.J.: Weighted spanning tree constraint with explanations. In: Quimper, C.-G. (ed.) CPAIOR 2016. LNCS, vol. 9676, pp. 98–107. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-33954-2_8
Delaite, A., Pesant, G.: Counting weighted spanning trees to solve constrained minimum spanning tree problems. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 176–184. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59776-8_14
Dooms, G., Katriel, I.: The Minimum Spanning Tree constraint. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 152–166. Springer, Heidelberg (2006). https://doi.org/10.1007/11889205_13
Dooms, G., Katriel, I.: The “Not-Too-Heavy Spanning Tree” constraint. In: Van Hentenryck, P., Wolsey, L. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 59–70. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72397-4_5
Ducomman, S., Cambazard, H., Penz, B.: Alternative filtering for the weighted circuit constraint: comparing lower bounds for the TSP and solving TSPTW. In: Schuurmans and Wellman [22], pp. 3390–3396
Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. 69B, 125–130 (1965)
Focacci, F., Lodi, A., Milano, M.: Cost-based domain filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-540-48085-3_14
Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)
Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees: Part II. Math. Program. 1, 6–25 (1970)
Kilby, P., Shaw, P.: Vehicle routing. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming. Foundations of Artificial Intelligence, vol. 2, pp. 801–836. Elsevier, New York (2006)
Pesant, G.: Counting-Based Search for Constraint Optimization Problems. In: Schuurmans and Wellman [22], pp. 3441–3448
Pesant, G., Quimper, C.-G., Zanarini, A.: Counting-based search: branching heuristics for constraint satisfaction problems. J. Artif. Int. Res. 43(1), 173–210 (2012)
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). https://doi.org/10.1007/978-3-540-68155-7_19
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). https://doi.org/10.1007/978-3-642-13520-0_31
Schuurmans, D., Wellman, M.P. (eds.): Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 12–17 February 2016. AAAI Press, Phoenix (2016)
Soules, G.W.: New permanental upper bounds for nonnegative matrices. Linear Multilinear A. 51(4), 319–337 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Coste, P., Lodi, A., Pesant, G. (2019). Using Cost-Based Solution Densities from TSP Relaxations to Solve Routing Problems. In: Rousseau, LM., Stergiou, K. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019. Lecture Notes in Computer Science(), vol 11494. Springer, Cham. https://doi.org/10.1007/978-3-030-19212-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-19212-9_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19211-2
Online ISBN: 978-3-030-19212-9
eBook Packages: Computer ScienceComputer Science (R0)