The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. According to the property of the demand of customers, we distinguish three variants of CVRP: unit-demand, splittable and unsplittable. We consider $k$-CVRP in general metrics and general graphs, where $k$ is the capacity of the vehicle and all the three versions are APX-hard for each fixed $k\geq 3$. In this paper, we give a $(5/2-\Theta(\sqrt{1/k}))$-approximation algorithm for splittable and unit-demand $k$-CVRP and a $(5/2+\ln2-\Theta(\sqrt{1/k}))$-approximation algorithm for unsplittable $k$-CVRP. Our approximation ratio is better than all previous results for $k$ smaller than a sufficiently large value, say $k\leq 1.7\times 10^7$. For small $k$, we also design independent elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the ratio from $1.792$ to $1.500$ for $k=3$, and from $1.750$ to $1.500$ for $k=4$ too. For the unsplittable case, we improve the 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 $k=3$ also surprisingly achieve the same ratio for the splittable case. Note that for small $k$ such as $3$, $4$ and $5$, some previous results have also been kept for decades. Our techniques, such as the EX-ITP method -- an extension of the classic ITP method, has potential to improve algorithms for more routing problems.

翻译：容量车辆路径问题（CVRP）是组合优化中研究最广泛的问题之一。根据客户需求的性质，我们区分了CVRP的三种变体：单位需求、可分需求和不可分需求。我们考虑一般度量空间和一般图中的$k$-CVRP，其中$k$是车辆容量，且对于每个固定的$k\geq 3$，这三个版本都是APX-难的。在本文中，我们针对可分需求和单位需求的$k$-CVRP给出了一个$(5/2-\Theta(\sqrt{1/k}))$-近似算法，针对不可分需求的$k$-CVRP给出了一个$(5/2+\ln2-\Theta(\sqrt{1/k}))$-近似算法。对于小于足够大值（例如$k\leq 1.7\times 10^7$）的$k$，我们的近似比优于以往所有结果。对于较小的$k$，我们还设计了独立的优雅算法，并取得了进一步改进。对于可分需求和单位需求情形，当$k=3$时，我们将比率从$1.792$改进至$1.500$；当$k=4$时，从$1.750$改进至$1.500$。对于不可分需求情形，当$k=3$时，我们将比率从$1.792$改进至$1.500$；当$k=4$时，从$2.051$改进至$1.750$；当$k=5$时，从$2.249$改进至$2.157$。值得注意的是，$k=3$时的近似比与可分情形下的比率相同。注意，对于$k=3$、$4$和$5$等小容量值，部分先前结果已保持数十年不变。我们的技术，例如EX-ITP方法（即经典ITP方法的扩展），有望改进更多路径规划问题的算法。