In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is \(\mathcal{NP}\)-complete, even when the underlying metric is induced by a weighted star. We present a 4-approximation for this problem on tree metrics. On general metrics, we obtain an O(logD) approximation algorithm, and also an \((O(\log \frac{1}{\epsilon}),1+\epsilon)\) bicriteria approximation. All these algorithms have running times that are almost linear in the input size. On instances that have an optimal solution with one r-path, we show how to obtain in polynomial time, a solution using at most 14 r-paths.

We also consider a linear relaxation for this problem that can be solved approximately using techniques of Carr & Vempala [7]. We obtain upper bounds on the integrality gap of this relaxation both in tree metrics and in general.


Vehicle Route Problem Dual Solution Vehicle Route General Metrics Approximation Guarantee 
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  1. 1.
    Arkin, E.M., Hassin, R., Levin, A.: Approximations for Minimum and Min-max Vehicle Routing Problems. Journal of Algorithms (2005)Google Scholar
  2. 2.
    Arkin, E.M., Mitchell, J.S.B., Narasimhan, G.: Resource-constrained Geometric Network Optimization. In: SCG 1998: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, pp. 307–316 (1998)Google Scholar
  3. 3.
    Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation Algorithms for Deadline-TSP and Vehicle Routing with Time Windows. In: Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing, pp. 166–174 (2004)Google Scholar
  4. 4.
    Bar-Yehuda, R., Even, G., Shahar, S.M.: On Approximating a Geometric Prize-Collecting Traveling Salesman Problem with Time Windows. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 55–66. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Bazgan, C., Hassin, R., Monnot, J.: Approximation Algorithms for Some Vehicle Routing Problems. Discrete Applied Mathematics 146, 27–42 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blum, A., Chawla, S., Karger, D.R., Lane, T., Meyerson, A., Minkoff, M.: Approximation Algorithms for Orienteering and Discounted-Reward TSP. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 46–55 (2003)Google Scholar
  7. 7.
    Carr, B., Vempala, S.: Randomized meta-rounding. In: 32nd ACM Symposium on the Theory of Computing, pp. 58–62 (2000)Google Scholar
  8. 8.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. GSIA, CMU-Report 388 (1977)Google Scholar
  9. 9.
    Li, C.L., Simchi-Levi, D., Desrochers, M.: On the distance constrained vehicle routing problem. Operations Research 40, 790–799 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Desrochers, M., Desrosiers, J., Solomon, M.: A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows. Operation Research 40, 342–354 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kantor, M., Rosenwein, M.: The Orienteering Problem with Time Windows. Journal of the Operational Research Society 43, 629–635 (1992)zbMATHCrossRefGoogle Scholar
  12. 12.
    Khuller, S., Malekian, A., Mestre, J.: To Fill or not to Fill: The Gas Station Problem (manuscript, 2006)Google Scholar
  13. 13.
    Kohen, A., Kan, A.R., Trienekens, H.: Vehicle Routing with Time Windows. Operations Research 36, 266–273 (1987)Google Scholar
  14. 14.
    Nagarajan, V., Ravi, R.: Minimum Vehicle Routing with a Common Deadline (2006),
  15. 15.
    Savelsbergh, M.: Local Search for Routing Problems with Time Windows. Annals of Operations Research 4, 285–305 (1985)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tan, K.C., Lee, L.H., Zhu, K.Q., Ou, K.: Heuristic Methods for Vehicle Routing Problems with Time Windows. Artificial Intelligence in Engineering, 281–295 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Viswanath Nagarajan
    • 1
  • R. Ravi
    • 1
  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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