Advertisement

Abstract

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.

Keywords

Vehicle Route Problem Dual Solution Vehicle Route General Metrics Approximation Guarantee 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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), https://server1.tepper.cmu.edu/gsiadoc/WP/2006-E53.pdf
  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

Personalised recommendations