Abstract
The Time Dependent Traveling Salesman Problem (TDTSP) is a generalization of the classical Traveling Salesman Problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 60 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet defining cuts. In particular, we also show that some facet defining cuts for the usual Asymmetric TSP formulation define low dimensional faces of the TDTSP formulation and give a way to lift them. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving several instances of reasonable size at the root node.
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References
Picard, J., Queyranne, M.: The time-dependent traveling salesman problem and its application to the tardiness problem in one-machine scheduling. Operations Research 26, 86–110 (1978)
Vajda, S.: Mathematical Programming. Addison-Wesley, Reading (1961)
Pessoa, A., Poggi de Aragão, M., Uchoa, E.: Robust Branch-Cut-and-Price Algorithms for Vehicle Routing Problems. In: Golden, B., Raghavan, S., Wasil, E. (eds.) The Vehicle Routing Problem: Latest Advances and New Challenges, pp. 297–326. Springer, New York (2008)
Pessoa, A., Uchoa, E., Poggi de Aragão, M.: A robust branch-cut-and-price algorithm for the heterogeneous fleet vehicle routing problem. Networks 54, 167–177 (2009)
Pessoa, A., Uchoa, E., Poggi de Aragão, M., Freitas, R.: Algorithms over Arc-time Indexed Formulations for Single and Parallel Machine Scheduling Problems. Technical report RPEP 8(8), Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil (2008)
Balas, E., Fischetti, M.: Polyhedral theory for the ATSP. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and Its Variations, pp. 117–168. Kluwer, Dordrecht (2002)
Balas, E., Carr, R., Fischetti, M., Simonetti, N.: New Facets of the STS Polytope Generated from Known Facets of the ATS Polytope. Discrete Optimization 3, 3–19 (2006)
Gouveia, L., Simonetti, L., Uchoa, E.: Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. Mathematical Programming (2009) (Online first)
Groetschel, M., Padberg, M.W.: Polyhedral theory. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem, pp. 251–305. Wiley, Chichester (1985)
Miranda Bront, J.J., Méndez-Díaz, I., Zabala, P.: An Integer Programming Approach for the Time Dependent Traveling Saleman Problem. In: 23rd European Conference on Operational Research (2009)
Fox, K., Gavish, B., Graves, S.: An N-Constraint Formulation of the (Time Dependent) Traveling Salesman Problem. Operations Research 28, 1018–1021 (1980)
Gouveia, L., Voss, S.: A Classification of formulations for the (time-dependent) traveling salesman problem. European Journal of Operations Research 83, 69–82 (1995)
Vander Wiel, R.J., Sahinidis, N.V.: Heuristics Bounds and Test Problem Generation for the time-dependent traveling salesman problem. Transportation Science 29, 167–183 (1995)
Bigras, L.-P., Gamache, M., Savard, G.: The Time-Dependent Traveling Salesman Problem and Single Machine Scheduling Problems with Sequence Dependent Setup Time. Discrete Optimization 5, 685–699 (2008)
Vander Wiel, R.J., Sahinidis, N.V.: An Exact Solution Approach for the time-dependent traveling salesman problem. Naval Research Logistics 43, 797–820 (1996)
Fischetti, M., Laporte, G., Martello, S.: The Delivery Man Problem and Cumulative Matroids. Operations Research 41, 1055–1064 (1993)
Lucena, A.: Time-Dependent Traveling Salesman Problem - The Deliveryman Case. Networks 20, 753–763 (1990)
Méndez-Díaz, I., Zabala, P., Lucena, A.: A New Formulation for the Traveling Deliveryman Problem. Discrete Applied Mathematics 156, 3233–3237 (2008)
Bentner, J., Bauer, G., Obermair, G.M., Morgensten, I., Schneider, J.: Optimization of the time-dependent traveling salesman problem with Monte Carlo methods. Physical Review E 64, 36701-1–36701-8 (2001)
Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton Univerisity Press, Princeton (1962)
Gale, D.: A theorem of flows in networks. Pacific Journal of Mathematics 7, 1073–1082 (1957)
Hoffman, A.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proceedings of Symposium in Applied Mathematics, vol. 10, pp. 113–128 (1960)
Hall, P.: On representatives of subsets. Journal of London Mathematical Society 10, 26–30 (1935)
Dantzig, G.B., Fulkerson, D.R., Johnson, S.M.: Solution of a large-scale Traveling Salesman Problem. Operations Research 2, 393–410 (1954)
Grotschel, M., Padberg, M.: On the Symmetric Traveling Salesman Problem II: Lifing Theorems and Facets. Mathematical Programming 16, 281–302 (1979)
Padberg, M.: On the facial structure of set packing polyhedra. Mathematical Programming 5, 199–215 (1973)
Irnich, S., Villeneuve, D.: The shortest path problem with resource constraints and k-cycle elimination for k ≥ 3. INFORMS Journal on Computing 18, 391–406 (2006)
Applegate, D., Bixby, R., Chvátal, V., Cook, W.: On The Solution Of Traveling Salesman Problems. Documenta Mathematica (Extra vol. ICM 3), 645–646 (1998)
Niskanen, S., Ostergard, P.R.J.: Cliquer user’s guide. Helsinki University of Technology, Communications Laboratory, Technical report 48 (2003)
Ralphs, T.K., Ladányi, L.: COIN/BCP User’s Manual (2001), http://www.coin-or.org/Presentations/bcp-man.pdf
Reinelt, G.: TSPLIB - A traveling salesman problem library. ORSA Journal on Computing 3(4), 376–384 (1991)
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Abeledo, H., Fukasawa, R., Pessoa, A., Uchoa, E. (2010). The Time Dependent Traveling Salesman Problem: Polyhedra and Branch-Cut-and-Price Algorithm. In: Festa, P. (eds) Experimental Algorithms. SEA 2010. Lecture Notes in Computer Science, vol 6049. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13193-6_18
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DOI: https://doi.org/10.1007/978-3-642-13193-6_18
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