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Probabilistic Analysis of Euclidean Capacitated Vehicle Routing

Authors Claire Mathieu, Hang Zhou

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Author Details

Claire Mathieu
  • CNRS, IRIF, Université de Paris, France
Hang Zhou
  • École Polytechnique, Institut Polytechnique de Paris, France

Funding

This work was partially funded by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR).

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Claire Mathieu and Hang Zhou. Probabilistic Analysis of Euclidean Capacitated Vehicle Routing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.43

Abstract

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(√n) or ω(√n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = √n, the ITP algorithm is at best a (1+c₀)-approximation for some positive constant c₀. On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+α due to Bompadre, Dror, and Orlin, where α is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+α. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • capacitated vehicle routing
  • iterated tour partitioning
  • probabilistic analysis
  • approximation algorithms

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References

  1. A. Adamaszek, A. Czumaj, and A. Lingas. PTAS for k-tour cover problem on the plane for moderately large values of k. International Journal of Foundations of Computer Science, 21(06):893-904, 2010. Google Scholar
  2. K. Altinkemer and B. Gavish. Heuristics for delivery problems with constant error guarantees. Transportation Science, 24(4):294-297, 1990. Google Scholar
  3. S. P. Anbuudayasankar, K. Ganesh, and S. Mohapatra. Models for practical routing problems in logistics. Springer, 2016. Google Scholar
  4. S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM (JACM), 45(5):753-782, 1998. Google Scholar
  5. T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama. 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, pages 275-283, 1997. Google Scholar
  6. A. Baltz, D. Dubhashi, A. Srivastav, L. Tansini, and S. Werth. Probabilistic analysis for a multiple depot vehicle routing problem. Random Structures & Algorithms, 30(1-2):206-225, 2007. Google Scholar
  7. J. Beardwood, J. H. Halton, and J. M. Hammersley. The shortest path through many points. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 55, pages 299-327. Cambridge University Press, 1959. Google Scholar
  8. J. Blauth, V. Traub, and J. Vygen. Improving the approximation ratio for capacitated vehicle routing. In Integer Programming and Combinatorial Optimization (IPCO), volume 12707. Springer, 2021. Google Scholar
  9. A. Bompadre, M. Dror, and J. B. Orlin. Improved bounds for vehicle routing solutions. Discrete Optimization, 3(4):299-316, 2006. Google Scholar
  10. A. Bompadre, M. Dror, and J. B. Orlin. Probabilistic analysis of unit-demand vehicle routeing problems. Journal of applied probability, 44(1):259-278, 2007. Google Scholar
  11. T. G. Crainic and G. Laporte. Fleet management and logistics. Springer Science & Business Media, 2012. Google Scholar
  12. C. F. Daganzo. The distance traveled to visit n points with a maximum of c stops per vehicle: An analytic model and an application. Transportation science, 18(4):331-350, 1984. Google Scholar
  13. A. Das and C. Mathieu. A quasi-polynomial time approximation scheme for Euclidean capacitated vehicle routing. Algorithmica, 73(1):115-142, 2015. Google Scholar
  14. B. Golden, S. Raghavan, and E. Wasil. The vehicle routing problem: latest advances and new challenges, volume 43 of Operations Research/Computer Science Interfaces Series. Springer, 2008. Google Scholar
  15. M. Haimovich and A. H. G. Rinnooy Kan. Bounds and heuristics for capacitated routing problems. Mathematics of operations Research, 10(4):527-542, 1985. Google Scholar
  16. Richard M. Karp. Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Mathematics of operations research, 2(3):209-224, 1977. Google Scholar
  17. M. Khachay and R. Dubinin. PTAS for the Euclidean capacitated vehicle routing problem in ℝ^d. In International Conference on Discrete Optimization and Operations Research, pages 193-205. Springer, 2016. Google Scholar
  18. C. L. Li and D. Simchi-Levi. Worst-case analysis of heuristics for multidepot capacitated vehicle routing problems. ORSA Journal on Computing, 2(1):64-73, 1990. Google Scholar
  19. C. L. Li, D. Simchi-Levi, and M. Desrochers. On the distance constrained vehicle routing problem. Operations research, 40(4):790-799, 1992. Google Scholar
  20. J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on computing, 28(4):1298-1309, 1999. Google Scholar
  21. G. Mosheiov. Vehicle routing with pick-up and delivery: tour-partitioning heuristics. Computers & Industrial Engineering, 34(3):669-684, 1998. Google Scholar
  22. M. D. Penrose and J. E. Yukich. Weak laws of large numbers in geometric probability. The Annals of Applied Probability, 13(1):277-303, 2003. Google Scholar
  23. W. T. Rhee. Probabilistic analysis of a capacitated vehicle routing problem II. The Annals of Applied Probability, 4(3):741-764, 1994. Google Scholar
  24. S. Steinerberger. New bounds for the traveling salesman constant. Advances in Applied Probability, 47(1):27-36, 2015. Google Scholar
  25. P. Toth and D. Vigo. The Vehicle Routing Problem. Society for Industrial and Applied Mathematics, 2002. Google Scholar
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