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Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

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  • Xu, Liang
  • Xu, Zhou
  • Xu, Dongsheng

Abstract

Given k identical salesmen, where k⩾2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k=2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k⩾2, closing a question that has remained open for a decade. Along with this, we have further developed a (1+ϵ)-approximation algorithm for any ϵ>0.

Suggested Citation

  • Xu, Liang & Xu, Zhou & Xu, Dongsheng, 2013. "Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree," European Journal of Operational Research, Elsevier, vol. 227(2), pages 284-292.
    • Handle: RePEc:eee:ejores:v:227:y:2013:i:2:p:284-292
    DOI: 10.1016/j.ejor.2012.12.023
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    References listed on IDEAS

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    1. D. S. Johnson & K. A. Niemi, 1983. "On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 1-14, February.
    2. Rego, César & Gamboa, Dorabela & Glover, Fred & Osterman, Colin, 2011. "Traveling salesman problem heuristics: Leading methods, implementations and latest advances," European Journal of Operational Research, Elsevier, vol. 211(3), pages 427-441, June.
    3. Martine Labbé & Gilbert Laporte & Hélène Mercure, 1991. "Capacitated Vehicle Routing on Trees," Operations Research, INFORMS, vol. 39(4), pages 616-622, August.
    4. van der Merwe, D.J. & Hattingh, J.M., 2006. "Tree knapsack approaches for local access network design," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1968-1978, November.
    5. Laporte, Gilbert, 1992. "The traveling salesman problem: An overview of exact and approximate algorithms," European Journal of Operational Research, Elsevier, vol. 59(2), pages 231-247, June.
    6. Pureza, Vitória & Morabito, Reinaldo & Reimann, Marc, 2012. "Vehicle routing with multiple deliverymen: Modeling and heuristic approaches for the VRPTW," European Journal of Operational Research, Elsevier, vol. 218(3), pages 636-647.
    7. Tetsuo Asano & Naoki Katoh & Kazuhiro Kawashima, 2001. "A New Approximation Algorithm for the Capacitated Vehicle Routing Problem on a Tree," Journal of Combinatorial Optimization, Springer, vol. 5(2), pages 213-231, June.
    8. Igor Averbakh & Oded Berman, 2002. "Minmax p-Traveling Salesmen Location Problems on a Tree," Annals of Operations Research, Springer, vol. 110(1), pages 55-68, February.
    9. Bektas, Tolga, 2006. "The multiple traveling salesman problem: an overview of formulations and solution procedures," Omega, Elsevier, vol. 34(3), pages 209-219, June.
    10. Goel, Asvin & Gruhn, Volker, 2008. "A General Vehicle Routing Problem," European Journal of Operational Research, Elsevier, vol. 191(3), pages 650-660, December.
    11. Refael Hassin, 1992. "Approximation Schemes for the Restricted Shortest Path Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 36-42, February.
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