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arXiv · 2210.16534

Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity

Jingyang Zhao·Mingyu Xiao
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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 $k\geq3$. Assume that the approximation ratio of metric TSP is $\frac{3}{2}$. We present a $(\frac{5}{2}-Θ(\frac{1}{\sqrt{k}}))$-approximation algorithm for the splittable and unit-demand cases, and a $(\frac{5}{2}+\ln2-Θ(\frac{1}{\sqrt{k}}))$-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 $1.7\times10^7$. 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, Mingyu Xiao. 2025-07-04. Improved Approximation Algorithms for Capacitated Vehicle Routing with Fixed Capacity. https://arxiv.org/abs/2210.16534

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