2026-07-22T16:12:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15476 2024-03-06T10:55:23Z ddc:004 open_access

Probabilistic Analysis of Euclidean Capacitated Vehicle Routing Mathieu, Claire Zhou, Hang capacitated vehicle routing iterated tour partitioning probabilistic analysis approximation algorithms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Claire Mathieu and Hang Zhou 2021 Is Part Of LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2021.43 urn:nbn:de:0030-drops-154769 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.43 eng https://creativecommons.org/licenses/by/4.0/legalcode