An Extension of the Das and Mathieu QPTAS to the Case of Polylog Capacity Constrained CVRP in Metric Spaces of a Fixed Doubling Dimension

  • Michael KhachayEmail author
  • Yuri Ogorodnikov
  • Daniel Khachay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


The Capacitated Vehicle Routing Problem (CVRP) is the well-known combinatorial optimization problem having numerous practically important applications. CVRP is strongly NP-hard (even on the Euclidean plane), hard to approximate in the general case and APX-complete for an arbitrary metric. Meanwhile, for the geometric settings of the problem, there are known a number of quasi-polynomial and even polynomial time approximation schemes. Among these results, the well-known QPTAS proposed by A. Das and C. Mathieu appears to be the most general. In this paper, we propose the first extension of this scheme to a more wide class of metric spaces. Actually, we show that the metric CVRP has a QPTAS any time when the problem is set up in the metric space of any fixed doubling dimension \(d>1\) and the capacity does not exceed \(\mathrm {polylog}\,{(}n)\).


  1. 1.
    Abraham, I., Bartal, Y., Neiman, O.: Advances in metric embedding theory. Adv. Math. 228(6), 3026–3126 (2011). Scholar
  2. 2.
    Adamaszek, A., Czumaj, A., Lingas, A.: PTAS for k-tour cover problem on the plane rof moderately large values of \(k\). Int. J. Found. Comput. Sci. 21(6), 893–904 (2010). Scholar
  3. 3.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 753–782 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45, 70–122 (1998). Scholar
  5. 5.
    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).
  6. 6.
    Avdoshin, S., Beresneva, E.: Local search metaheuristics for capacitated vehicle routing problem: a comparative study. Proc. Inst. Syst. Program. RAS 31, 121–138 (2019). Scholar
  7. 7.
    Bartal, Y., Gottlieb, L.A., Krauthgamer, R.: The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme. SIAM J. Comput. 45(4), 1563–1581 (2016). Scholar
  8. 8.
    Becker, A., Klein, P.N., Schild, A.: A PTAS for bounded-capacity vehicle routing in planar graphs. In: Friggstad, Z., Sack, J.-R., Salavatipour, M.R. (eds.) WADS 2019. LNCS, vol. 11646, pp. 99–111. Springer, Cham (2019). Scholar
  9. 9.
    Chen, J., Gui, P., Ding, T., Zhou, Y.: Optimization of transportation routing problem for fresh food by improved ant colony algorithm based on Tabu search. Sustainability 11 (2019).
  10. 10.
    Dantzig, G.B., Ramser, J.H.: The truck dispatching problem. Manage. Sci. 6(1), 80–91 (1959)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Das, A., Mathieu, C.: A quasipolynomial time approximation scheme for Euclidean capacitated vehicle routing. Algorithmica 73(1), 115–142 (2014). Scholar
  12. 12.
    Demir, E., Huckle, K., Syntetos, A., Lahy, A., Wilson, M.: Vehicle routing problem: past and future. In: Wells, P. (ed.) Contemporary Operations and Logistics, pp. 97–117. Springer, Cham (2019). Scholar
  13. 13.
    Frifita, S., Masmoudi, M.: VNS methods for home care routing and scheduling problem with temporal dependencies, and multiple structures and specialties. Int. Trans. Oper. Res. 27(1), 291–313 (2020). Scholar
  14. 14.
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: 44th Annual IEEE Symposium on Foundations of Computer Science 2003, Proceedings, pp. 534–543 (2003).
  15. 15.
    Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Math. Oper. Res. 10(4), 527–542 (1985). Scholar
  16. 16.
    Hokama, P., Miyazawa, F.K., Xavier, E.C.: A branch-and-cut approach for the vehicle routing problem with loading constraints. Expert Syst. Appl. 47, 1–13 (2016). Scholar
  17. 17.
    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
  18. 18.
    Khachai, M., Ogorodnikov, Y.: Haimovich–Rinnooy Kan polynomial-time approximation scheme for the CVRP in metric spaces of a fixed doubling dimension. Trudy instituta matematiki i mekhaniki UrO RAN 25(4), 235–248 (2019).
  19. 19.
    Khachai, M.Y., Ogorodnikov, Y.Y.: Polynomial-time approximation scheme for the capacitated vehicle routing problem with time windows. Proc. Steklov Inst. Math. 307(1), 51–63 (2019). Scholar
  20. 20.
    Khachay, M., Ogorodnikov, Y.: Efficient PTAS for the Euclidean CVRP with time windows. In: van der Aalst, W.M.P., et al. (eds.) AIST 2018. LNCS, vol. 11179, pp. 318–328. Springer, Cham (2018). Scholar
  21. 21.
    Khachay, M., Ogorodnikov, Y.: Approximation scheme for the capacitated vehicle routing problem with time windows and non-uniform demand. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 309–327. Springer, Cham (2019). Scholar
  22. 22.
    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
  23. 23.
    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
  24. 24.
    Nazari, M., Oroojlooy, A., Takáč, M., Snyder, L.V.: Reinforcement learning for solving the vehicle routing problem. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS 2018, pp. 9861–9871. Curran Associates Inc., Red Hook (2018)Google Scholar
  25. 25.
    Necula, R., Breaban, M., Raschip, M.: Tackling dynamic vehicle routing problem with time windows by means of ant colony system. In: 2017 IEEE Congress on Evolutionary Computation (CEC), pp. 2480–2487 (2017).
  26. 26.
    Papadimitriou, C.: Euclidean TSP is NP-complete. Theoret. Comput. Sci. 4, 237–244 (1977)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pessoa, A.A., Sadykov, R., Uchoa, E.: Enhanced branch-cut-and-price algorithm for heterogeneous fleet vehicle routing problems. Eur. J. Oper. Res. 270(2), 530–543 (2018). Scholar
  28. 28.
    Qiu, M., Fu, Z., Eglese, R., Tang, Q.: A tabu search algorithm for the vehicle routing problem with discrete split deliveries and pickups. Comput. Oper. Res. 100, 102–116 (2018). Scholar
  29. 29.
    Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC 2004, pp. 281–290. Association for Computing Machinery, New York (2004).
  30. 30.
    Toth, P., Vigo, D.: Vehicle Routing: Problems, Methods, and Applications. MOS-SIAM Series on Optimization, 2nd edn. SIAM, Philadelphia (2014)CrossRefGoogle Scholar
  31. 31.
    Vidal, T., Crainic, T.G., Gendreau, M., Prins, C.: A hybrid genetic algorithm with adaptive diversity management for a large class of vehicle routing problems with time windows. Comput. Oper. Res. 40(1), 475–489 (2013). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

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

Personalised recommendations