Finding the shortest travelling tour of vehicles with capacity k from the depot to the customers is called the Capacity vehicle routing problem (CVRP). CVRP plays an essential position in logistics systems, and it is the most intensively studied problem in combinatorial optimization. In complexity, CVRP with k $\ge$ 3 is an NP-hard problem, and it is APX-hard as well. We already knew that it could not be approximated in metric space. Moreover, it is the first problem resisting Arora's famous approximation framework. So, whether there is, a polynomial-time (1+$\epsilon$)-approximation for the Euclidean CVRP for any $\epsilon>0$ is still an open problem. This paper will summarize the research progress from history to up-to-date developments. The survey will be updated periodically.

翻译：从仓库到客户之间寻找容量为k的车辆的最短行驶路径被称为有容量约束的车辆路径问题（CVRP）。CVRP在物流系统中占据重要地位，也是组合优化中研究最为深入的问题之一。在复杂度方面，当k ≥ 3时，CVRP属于NP难问题，同时亦是APX难问题。已知在度量空间中该问题无法被近似求解，并且它是首个突破Arora著名近似框架的问题。因此，对于欧几里得空间中的CVRP，是否存在任意ε>0的多项式时间（1+ε）近似算法仍是一个开放问题。本文将系统梳理从历史至今的研究进展，并将定期更新综述内容。