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A Tight (1.5+ε)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees

Authors Claire Mathieu, Hang Zhou

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Author Details

Claire Mathieu
  • CNRS 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. A Tight (1.5+ε)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 91:1-91:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.91

Abstract

In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1. For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023]. In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time (1.5+ε)-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
Keywords
  • approximation algorithms
  • capacitated vehicle routing
  • graph algorithms
  • combinatorial optimization

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