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Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity

Authors Jingyang Zhao , Mingyu Xiao

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • Combinatorial Optimization
  • Capacitated Vehicle Routing
  • Approximation Algorithms
  • Graph Algorithms

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Abstract

The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Based on customer demand, we distinguish three variants of CVRP: unit-demand, splittable, and unsplittable. In this paper, we consider k-CVRP in general metrics and on general graphs, where k is the vehicle capacity. All three versions are APX-hard for any fixed k ≥ 3. Assume that the approximation ratio of metric TSP is 3/2. We present a (5/2 - Θ(√{1/k}))-approximation algorithm for the splittable and unit-demand cases, and a (5/2 + ln 2 - Θ(√{1/k}))-approximation algorithm for the unsplittable case. Our approximation ratio is better than the previous results when k is less than a sufficiently large value, approximately 1.7 x 10⁷.
For small values of k, we design independent and elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the approximation ratio from 1.792 to 1.500 for k = 3, and from 1.750 to 1.500 for k = 4. For the unsplittable case, we improve the approximation ratio from 1.792 to 1.500 for k = 3, from 2.051 to 1.750 for k = 4, and from 2.249 to 2.157 for k = 5. The approximation ratio for k = 3 surprisingly achieves the same value as in the splittable case. Our techniques, such as EX-ITP - an extension of the classic ITP method, have the potential to improve algorithms for other routing problems as well.

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Jingyang Zhao and Mingyu Xiao. Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.93

Author Details

Jingyang Zhao
  • University of Electronic Science and Technology of China, Chengdu, China
Mingyu Xiao
  • University of Electronic Science and Technology of China, Chengdu, China

Funding

The work is supported by the Postdoctoral Fellowship Program of CPSF under Grant Number GZC20251102, and the National Natural Science Foundation of China under the grants 62372095, 62172077, and 62350710215.

Acknowledgements

We would like to thank all reviewers for their valuable comments.

References

  1. Kemal Altinkemer and Bezalel Gavish. Heuristics for unequal weight delivery problems with a fixed error guarantee. Oper. Res. Lett., 6(4):149-158, 1987. Google Scholar
  2. Shoshana Anily and Julien Bramel. Approximation algorithms for the capacitated traveling salesman problem with pickups and deliveries. Naval Research Logistics, 46(6):654-670, 1999. Google Scholar
  3. Tetsuo Asano, Naoki Katoh, Hisao Tamaki, and Takeshi Tokuyama. Covering points in the plane by k-tours: a polynomial approximation scheme for fixed k. IBM Tokyo Research Laboratory Research Report RT0162, 1996. Google Scholar
  4. Tetsuo Asano, Naoki Katoh, Hisao Tamaki, and Takeshi Tokuyama. Covering points in the plane by k-tours: Towards a polynomial time approximation scheme for general k. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 275-283. ACM, 1997. URL: https://doi.org/10.1145/258533.258602.
  5. Cristina Bazgan, Refael Hassin, and Jérôme Monnot. Approximation algorithms for some vehicle routing problems. Discret. Appl. Math., 146(1):27-42, 2005. URL: https://doi.org/10.1016/j.dam.2004.07.003.
  6. Walter J Bell, Louis M Dalberto, Marshall L Fisher, Arnold J Greenfield, Ramchandran Jaikumar, Pradeep Kedia, Robert G Mack, and Paul J Prutzman. Improving the distribution of industrial gases with an on-line computerized routing and scheduling optimizer. Interfaces, 13(6):4-23, 1983. Google Scholar
  7. Jannis Blauth, Vera Traub, and Jens Vygen. Improving the approximation ratio for capacitated vehicle routing. Math. Program., 197(2):451-497, 2023. URL: https://doi.org/10.1007/s10107-022-01841-4.
  8. Agustín Bompadre, Moshe Dror, and James B. Orlin. Improved bounds for vehicle routing solutions. Discret. Optim., 3(4):299-316, 2006. URL: https://doi.org/10.1016/j.disopt.2006.04.002.
  9. Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Oper. Res. Forum, 3(1), 2022. URL: https://doi.org/10.1007/s43069-021-00101-z.
  10. Vasek Chvátal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4(3):233-235, 1979. URL: https://doi.org/10.1287/moor.4.3.233.
  11. Vincent Cohen-Addad, Arnold Filtser, Philip N. Klein, and Hung Le. On light spanners, low-treewidth embeddings and efficient traversing in minor-free graphs. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 589-600. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00061.
  12. George B Dantzig and John H Ramser. The truck dispatching problem. Manag. Sci., 6(1):80-91, 1959. Google Scholar
  13. Szymon Dudycz, Jan Marcinkowski, Katarzyna E. Paluch, and Bartosz Rybicki. A 4/5 - approximation algorithm for the maximum traveling salesman problem. In Friedrich Eisenbrand and Jochen Könemann, editors, Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, volume 10328 of Lecture Notes in Computer Science, pages 173-185. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_15.
  14. Arnold Filtser and Hung Le. Low treewidth embeddings of planar and minor-free metrics. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 1081-1092. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00105.
  15. Zachary Friggstad, Ramin Mousavi, Mirmahdi Rahgoshay, and Mohammad R. Salavatipour. Improved approximations for capacitated vehicle routing with unsplittable client demands. In Karen I. Aardal and Laura Sanità, editors, Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022, Eindhoven, The Netherlands, June 27-29, 2022, Proceedings, volume 13265 of Lecture Notes in Computer Science, pages 251-261. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06901-7_19.
  16. Harold Neil Gabow. Implementation of algorithms for maximum matching on nonbipartite graphs. PhD thesis, Stanford University, 1974. Google Scholar
  17. Michel X. Goemans and David P. Williamson. A general approximation technique for constrained forest problems. SIAM J. Comput., 24(2):296-317, 1995. URL: https://doi.org/10.1137/S0097539793242618.
  18. Anupam Gupta, Euiwoong Lee, and Jason Li. A local search-based approach for set covering. In Telikepalli Kavitha and Kurt Mehlhorn, editors, 2023 Symposium on Simplicity in Algorithms, SOSA 2023, Florence, Italy, January 23-25, 2023, pages 1-11. SIAM, SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977585.ch1.
  19. Mordecai Haimovich and Alexander H. G. Rinnooy Kan. Bounds and heuristics for capacitated routing problems. Math. Oper. Res., 10(4):527-542, 1985. URL: https://doi.org/10.1287/moor.10.4.527.
  20. David Hartvigsen. Extensions of matching theory. PhD thesis, Carnegie-Mellon University, 1984. Google Scholar
  21. Refael Hassin and Asaf Levin. A better-than-greedy approximation algorithm for the minimum set cover problem. SIAM J. Comput., 35(1):189-200, 2005. URL: https://doi.org/10.1137/S0097539704444750.
  22. Refael Hassin and Shlomi Rubinstein. An approximation algorithm for maximum packing of 3-edge paths. Inf. Process. Lett., 63(2):63-67, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00097-5.
  23. Refael Hassin and Shlomi Rubinstein. Better approximations for max TSP. Inf. Process. Lett., 75(4):181-186, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00097-1.
  24. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC 2021: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 32-45. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451009.
  25. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A deterministic better-than-3/2 approximation algorithm for metric TSP. In Alberto Del Pia and Volker Kaibel, editors, Integer Programming and Combinatorial Optimization - 24th International Conference, IPCO 2023, Madison, WI, USA, June 21-23, 2023, Proceedings, volume 13904 of Lecture Notes in Computer Science, pages 261-274. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32726-1_19.
  26. David G. Kirkpatrick and Pavol Hell. On the completeness of a generalized matching problem. In Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 240-245. ACM, 1978. URL: https://doi.org/10.1145/800133.804353.
  27. Eugene Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, 1976. Google Scholar
  28. Chung-Lun Li and David Simchi-Levi. Worst-case analysis of heuristics for multidepot capacitated vehicle routing problems. INFORMS J. Comput., 2(1):64-73, 1990. URL: https://doi.org/10.1287/ijoc.2.1.64.
  29. Claire Mathieu and Hang Zhou. Probabilistic analysis of euclidean capacitated vehicle routing. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 43:1-43:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.43.
  30. Claire Mathieu and Hang Zhou. A PTAS for capacitated vehicle routing on trees. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 95:1-95:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.95.
  31. Claire Mathieu and Hang Zhou. A tight (1.5+ε)-approximation for unsplittable capacitated vehicle routing on trees. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 91:1-91:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.91.
  32. Seth Pettie and Vijaya Ramachandran. An optimal minimum spanning tree algorithm. J. ACM, 49(1):16-34, 2002. URL: https://doi.org/10.1145/505241.505243.
  33. Robert Clay Prim. Shortest connection networks and some generalizations. Bell Syst. Tech. J., 36(6):1389-1401, 1957. Google Scholar
  34. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  35. Anatolii Ivanovich Serdyukov. Some extremal bypasses in graphs. Upravlyaemye Sistemy, 17:76-79, 1978. Google Scholar
  36. Paolo Toth and Daniele Vigo. Vehicle routing: problems, methods, and applications. SIAM, 2014. Google Scholar
  37. Eduardo Uchoa, Diego Pecin, Artur Alves Pessoa, Marcus Poggi, Thibaut Vidal, and Anand Subramanian. New benchmark instances for the capacitated vehicle routing problem. Eur. J. Oper. Res., 257(3):845-858, 2017. URL: https://doi.org/10.1016/j.ejor.2016.08.012.
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