The Capacitated Vehicle Routing Problem (CVRP) is a core NP-hard problem in the field of combinatorial optimization. It aims to plan optimal routes for a fleet of vehicles with uniform capacity, serving a set of customers with specific demands from a single depot, while minimizing the total travel distance. Due to its extensive applications in logistics, distribution, and supply chain management, CVRP has attracted significant research attention. Theoretically, the problem has been proven to be APX-hard, and in general metric spaces, approximate solutions of arbitrary precision cannot be obtained unless P=NP. These inherent complexities highlight the importance of developing approximation algorithms-finding solutions with provable performance guarantees in polynomial time. This paper aims to provide a systematic and comprehensive survey of the research progress on CVRP approximation algorithms. The paper first strictly defines CVRP and its key variants, and elucidates the sources of its fundamental computational complexity. Subsequently, the article deeply analyzes the core algorithmic frameworks and technical schools of thought in this field, including: the Iterated Tour Partitioning (ITP) framework that laid the foundation of the field and its latest improvements; the evolution of approximation schemes (PTAS/QPTAS) for geometric spaces such as Euclidean space; and modern algorithms for structured graphs like trees, planar graphs, and graphs with bounded highway dimension, with a particular focus on techniques based on metric embedding and linear programming relaxation. Finally, this paper summarizes the current best approximation ratios for various problem settings and systematically outlines the core unresolved open problems in the field, pointing out directions for future research.

翻译：容量受限车辆路径问题（Capacitated Vehicle Routing Problem, CVRP）是组合优化领域的核心NP-hard问题，旨在为车队统一容量的车辆规划最优路径，从单一仓库出发，服务一组具有特定需求的客户，同时最小化总行驶距离。由于其在物流、配送和供应链管理中的广泛应用，CVRP已吸引了大量研究关注。从理论上讲，该问题已被证明是APX-hard的，在一般度量空间中，除非P=NP，否则无法获得任意精度的近似解。这些固有复杂性凸显了开发近似算法的重要性——即在多项式时间内找到具有可证明性能保证的解。本文旨在系统且全面地综述CVRP近似算法的研究进展。文章首先严格定义CVRP及其关键变体，并阐明其基本计算复杂性的来源。随后，文章深入分析了该领域的核心算法框架和技术流派，包括：奠定领域基础的迭代划分路径（Iterated Tour Partitioning, ITP）框架及其最新改进；几何空间（如欧氏空间）中近似方案（PTAS/QPTAS）的演进；以及针对树、平面图和具有有界公路维数图等结构化图的现代算法，特别关注基于度量嵌入和线性规划松弛的技术。最后，本文总结了各种问题设定下的当前最佳近似比，并系统性地勾勒了该领域核心的未解决开放问题，为未来研究指明方向。