Algorithmica

, Volume 67, Issue 1, pp 49–64 | Cite as

The School Bus Problem on Trees

  • Adrian Bock
  • Elyot Grant
  • Jochen Könemann
  • Laura Sanità
Article

Abstract

The School Bus Problem is an NP-hard vehicle routing problem in which the goal is to route buses that transport children to a school such that for each child, the distance travelled on the bus does not exceed the shortest distance from the child’s home to the school by more than a given regret threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required.

In this paper, we give a polynomial time 4-approximation algorithm when the children and school are located at vertices of a fixed tree. As a byproduct of our analysis, we show that the integrality gap of the natural set-cover formulation for this problem is also bounded by 4. We also present a constant factor approximation for the variant where we have a fixed number of buses to use, and the goal is to minimize the maximum regret.

Keywords

Vehicle routing Approximation algorithm Set-cover formulation 

References

  1. 1.
    Adamaszek, A., Czumaj, A., Lingas, A.: PTAS for k-tour cover problem on the plane for moderately large values of k. In: Algorithms and Computation. LNCS, vol. 5878, pp. 994–1003. Springer, Berlin (2009) CrossRefGoogle Scholar
  2. 2.
    Asano, T., Katoh, N., Tamaki, H., Tokuyama, T.: Covering points in the plane by k-tours: towards a polynomial time approximation scheme for general k. STOC (1997) 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 36th Annual ACM Symposium on Theory of Computing, pp. 166–174 (2004) Google Scholar
  4. 4.
    Blum, A., Chawla, S., Karger, D.R., Lane, T., Meyerson, A., Minkoff, M.: Approximation algorithms for orienteering and discounted-reward TSP. SIAM J. Comput. 37(2), 653–670 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chekuri, C., Korula, N., Pal, M.: Improved algorithms for orienteering and related problems. In: SODA, pp. 661–670 (2008) Google Scholar
  6. 6.
    Desrochers, M., Desrosiers, J., Solomon, M.: A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40, 342–354 (1992) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Haimovich, M., Rinnoy Kan, A.H.G.: Bounds and heuristic for capacitated routing problems. Math. Oper. Res. 10(4), 527–542 (1985) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Labbe, M., Laporte, G., Mercure, H.: Capacitated vehicle routing on trees. Oper. Res. 39(4), 616–622 (1991) MATHCrossRefGoogle Scholar
  9. 9.
    Laporte, G., Desrochers, M., Norbert, Y.: Two exact algorithms for the distance constrained vehicle routing problem. Networks 14, 47–61 (1984) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, C.-L., Simchi-Levi, S., Desrochers, M.: On the distance constrained vehicle routing problem. Oper. Res. 40, 790–799 (1992) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Nagarajan, V., Ravi, R.: Approximation Algorithms for Distance Constrained Vehicle Routing Problems. Tepper School of Business, Carnegie–Mellon University Press, Pittsburgh (2008) Google Scholar
  12. 12.
    Park, J., Kim, B.-I.: The school bus routing problem: a review. Eur. J. Oper. Res. 202, 311–319 (2010) MATHCrossRefGoogle Scholar
  13. 13.
    Spada, M., Bierlaire, M., Liebling, Th.M.: Decision-aiding methodology for the school bus routing and scheduling problem. Transp. Sci. 39(4), 477–490 (2005) CrossRefGoogle Scholar
  14. 14.
    Toth, P., Vigo, D.: The Vehicle Routing Problem. SIAM, Philadelphia (2001) Google Scholar
  15. 15.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001) Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Adrian Bock
    • 1
  • Elyot Grant
    • 2
  • Jochen Könemann
    • 2
  • Laura Sanità
    • 2
  1. 1.EPFLLausanneSwitzerland
  2. 2.University of WaterlooWaterlooCanada

Personalised recommendations