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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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