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An Approximation Algorithm for Distance-Constrained Vehicle Routing on Trees

Authors Marc Dufay, Claire Mathieu, Hang Zhou



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Author Details

Marc Dufay
  • École Polytechnique and IRIF, Palaiseau, France
Claire Mathieu
  • CNRS Paris, France
Hang Zhou
  • École Polytechnique, Palaiseau, France

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Marc Dufay, Claire Mathieu, and Hang Zhou. An Approximation Algorithm for Distance-Constrained Vehicle Routing on Trees. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.27

Abstract

In the Distance-constrained Vehicle Routing Problem (DVRP), we are given a graph with integer edge weights, a depot, a set of n terminals, and a distance constraint D. The goal is to find a minimum number of tours starting and ending at the depot such that those tours together cover all the terminals and the length of each tour is at most D. The DVRP on trees is of independent interest, because it is equivalent to the "virtual machine packing" problem on trees studied by Sindelar et al. [SPAA'11]. We design a simple and natural approximation algorithm for the tree DVRP, parameterized by ε > 0. We show that its approximation ratio is α + ε, where α ≈ 1.691, and in addition, that our analysis is essentially tight. The running time is polynomial in n and D. The approximation ratio improves on the ratio of 2 due to Nagarajan and Ravi [Networks'12]. The main novelty of this paper lies in the analysis of the algorithm. It relies on a reduction from the tree DVRP to the bounded space online bin packing problem via a new notion of "reduced length".

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • vehicle routing
  • distance constraint
  • approximation algorithms
  • trees

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References

  1. Arjang A. Assad. Modeling and implementation issues in vehicle routing. In Vehicle routing: Methods and studies, pages 7-45, 1988. Google Scholar
  2. Sean Barker, Timothy Wood, Prashant Shenoy, and Ramesh Sitaraman. An empirical study of memory sharing in virtual machines. In USENIX Annual Technical Conference, pages 273-284, 2012. Google Scholar
  3. Amariah Becker and Alice Paul. A framework for vehicle routing approximation schemes in trees. In Workshop on Algorithms and Data Structures, pages 112-125. Springer, 2019. Google Scholar
  4. Teodor G. Crainic and Gilbert Laporte. Fleet management and logistics. Springer Science & Business Media, 2012. Google Scholar
  5. Zachary Friggstad and Chaitanya Swamy. Approximation algorithms for regret-bounded vehicle routing and applications to distance-constrained vehicle routing. In Proceedings of the forty-sixth annual ACM Symposium on Theory of Computing (STOC), pages 744-753, 2014. Google Scholar
  6. Michael R. Garey and David S. Johnson. Computers and Intractability: a guide to the theory of NP-Completeness. Freeman, New York, 1985. Google Scholar
  7. Bruce Golden, S. Raghavan, and Edward Wasil. The vehicle routing problem: latest advances and new challenges, volume 43 of Operations Research/Computer Science Interfaces Series. Springer, 2008. Google Scholar
  8. Aditya Jayaprakash and Mohammad R. Salavatipour. Approximation schemes for capacitated vehicle routing on graphs of bounded treewidth, bounded doubling, or highway dimension. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 877-893, 2022. Google Scholar
  9. Gilbert Laporte, Martin Desrochers, and Yves Nobert. Two exact algorithms for the distance-constrained vehicle routing problem. Networks, 14(1):161-172, 1984. Google Scholar
  10. Chan C. Lee and Der-Tsai Lee. A simple on-line bin-packing algorithm. Journal of the ACM (JACM), 32(3):562-572, 1985. Google Scholar
  11. Chung-Lun Li, David Simchi-Levi, and Martin Desrochers. On the distance constrained vehicle routing problem. Operations Research, 40(4):790-799, 1992. Google Scholar
  12. Claire Mathieu and Hang Zhou. A PTAS for capacitated vehicle routing on trees. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), pages 95:1-95:20, 2022. Google Scholar
  13. Viswanath Nagarajan and R Ravi. Approximation algorithms for distance constrained vehicle routing problems. Networks, 59(2):209-214, 2012. Google Scholar
  14. Safraz Rampersaud and Daniel Grosu. Sharing-aware online virtual machine packing in heterogeneous resource clouds. IEEE Transactions on Parallel and Distributed Systems, 28(7):2046-2059, 2016. Google Scholar
  15. Michael Sindelar, Ramesh K. Sitaraman, and Prashant Shenoy. Sharing-aware algorithms for virtual machine colocation. In Proceedings of the twenty-third annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 367-378, 2011. Google Scholar
  16. Paolo Toth and Daniele Vigo. The Vehicle Routing Problem. SIAM, 2002. Google Scholar
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