Improved Polynomial Time Approximation Scheme for Capacitated Vehicle Routing Problem with Time Windows

  • Michael KhachayEmail author
  • Yuri Ogorodnikov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)


The Capacitated Vehicle Routing Problem with Time Windows is the well-known combinatorial optimization problem having numerous valuable applications in operations research. In this paper, following the famous framework by M. Haimovich and A. Rinnooy Kan and technique by T. Asano et al., we propose a novel approximation scheme for the planar Euclidean CVRPTW. For any fixed \(\varepsilon >0\), the proposed scheme finds a \((1+\varepsilon )\)-approximate solution of CVRPTW in time
$$TIME(\mathrm {TSP},\rho ,n)+O(n^2)+O\left( e^{O\left( q\,\left( \frac{q}{\varepsilon }\right) ^3(p\rho )^2\log (p\rho )\right) }\right) ,$$
where q is the given vehicle capacity bound, p is the number of time windows for servicing the customers, and \(TIME(\mathrm {TSP},\rho ,n)\) is the time needed to find a \(\rho \)-approximate solution for an auxiliary instance of the metric TSP.


Capacitated Vehicle Routing Problem Time windows Efficient Polynomial Time Approximation Scheme 


  1. 1.
    Andrews, G.E., Eriksson, K.: Integer Partitions, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 753–782 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC 1997, pp. 275–283. ACM, New York (1997).,
  4. 4.
    Dantzig, G.B., Ramser, J.H.: The truck dispatching problem. Manage. Sci. 6(1), 80–91 (1959)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Das, A., Mathieu, C.: A quasipolynomial time approximation scheme for Euclidean capacitated vehicle routing. Algorithmica 73, 115–142 (2015). Scholar
  6. 6.
    Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Math. Oper. Res. 10(4), 527–542 (1985). Scholar
  7. 7.
    Khachai, M.Y., Dubinin, R.D.: Approximability of the vehicle routing problem in finite-dimensional Euclidean spaces. Proc. Steklov Inst. Math. 297(1), 117–128 (2017). Scholar
  8. 8.
    Khachay, M., Ogorodnikov, Y.: Efficient PTAS for the Euclidean CVRP with time windows. In: Analysis of Images, Social Networks and Texts - 7th International Conference (AIST 2018). LNCS, vol. 11179, pp. 296–306 (2018). Scholar
  9. 9.
    Khachay, M., Dubinin, R.: PTAS for the Euclidean capacitated vehicle routing problem in \(R^d\). In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 193–205. Springer, Cham (2016). Scholar
  10. 10.
    Khachay, M., Zaytseva, H.: Polynomial time approximation scheme for single-depot Euclidean capacitated vehicle routing problem. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 178–190. Springer, Cham (2015). Scholar
  11. 11.
    Kumar, S., Panneerselvam, R.: A survey on the vehicle routing problem and its variants. Intell. Inf. Manage. 4, 66–74 (2012). Scholar
  12. 12.
    Song, L., Huang, H.: The Euclidean vehicle routing problem with multiple depots and time windows. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10628, pp. 449–456. Springer, Cham (2017). Scholar
  13. 13.
    Song, L., Huang, H., Du, H.: Approximation schemes for Euclidean vehicle routing problems with time windows. J. Comb. Optim. 32(4), 1217–1231 (2016). Scholar
  14. 14.
    Toth, P., Vigo, D.: Vehicle Routing: Problems, Methods, and Applications. MOS-SIAM Series on Optimization, 2nd edn. SIAM, Philadelphia (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Krasovsky Institute of Mathematics and MechanicsEkaterinburgRussia
  2. 2.Ural Federal UniversityEkaterinburgRussia
  3. 3.Omsk State Technical UniversityOmskRussia

Personalised recommendations