In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [14], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d_v and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed \(ε \gt \frac{1}{3000}\), where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [15] presented an approximation scheme with time n^logO(1/ε)n for Euclidean plane ℝ². No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n^O(log6n/ε5).

在本文中，我们介绍了针对几类图的容量车辆路径问题 (CVRP) 的近似方案。在由 Dantzig 和 Ramser 于 1959 年提出的 CVRP [14] 中，我们给出了一个具有度量边成本的图G=(V,E) 、一个车厂r ∈ V和一个有界容量Q的车辆。目标是为返回站点的车辆找到最小成本的旅行集合，每次访问最多Q个节点，以便它们覆盖所有节点。这概括了经典的 TSP 并得到了广泛的研究。在更一般的设置中，每个节点v都有一个需求d _v并且每次旅行的总需求必须不超过Q. 每个节点的需求要么必须由一个旅游服务（不可拆分），要么可以由多个旅游服务（可拆分）。对于一些固定的 \(ε \gt \frac{1}{3000 }\)，其中 α 是 TSP 的最佳近似值。即使对于树的情况，最佳近似比率也是 4/3 [5]，并且对于这种简单的图类是否存在近似方案一直是一个悬而未决的问题。Das 和 Mathieu [15] 提出了欧几里德平面 ℝ ^2的时间n ^{log O(1/ε) n}的近似方案。对于任何其他类别的度量（没有对Q 的进一步限制），没有其他近似方案是已知的). 在本文中，我们通过为有界树宽图、有界公路维数图和有界倍增维数图提出准多项式时间近似方案 (QPTAS)，在这个经典问题上取得了重大进展。为了进行比较，我们的结果暗示了运行时间为n ^{O(log 6 n/ε 5 )}的欧几里德平面的近似方案。