In this paper, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser (1959), we are given a graph $G=(V,E)$ with metric edges costs, a depot $r\in V$, and a vehicle of bounded capacity $Q$. The goal is to find 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 tour (splittable). The best known approximation algorithm for general graphs has ratio $\alpha+2(1-\epsilon)$ (for the unsplittable) and $\alpha+1-\epsilon$ (for the splittable) for some fixed $\epsilon>\frac{1}{3000}$, where $\alpha$ is the best approximation for TSP. Even for the case of trees, the best approximation ratio is $4/3$ by Becker (2018) and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu (2015) presented an approximation scheme with time $n^{\log^{O(1/\epsilon)}n}$ for Euclidean plane $\mathbb{R}^2$. No other approximation scheme is known for any other class of metrics (without further restrictions on $Q$). In this paper, 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 Euclidean plane with run time $n^{O(\log^{10}n/\epsilon^{9})}$.
翻译:本文中, 我们以多个图表类别 来展示“ 快速车辆运行比例 ” (CVRP) 。 在由 Dantzig 和 Ramser (1959年) 推出的 CVRP 中, 我们得到了一个具有度边缘成本的G=( V, E) 美元, 一个仓库美元, 和一个装有约束容量的车辆 Q美元。 目标是为返回仓库的车辆找到最低的旅游费用收集, 每个以 Q$ 的节点访问 。 这样可以覆盖所有的节点 。 这将经典的 TSP 并进行广泛研究。 在更普遍的设置中, 每笔美元都有 $d_ v美元, 每次巡航的总需求不能超过 Q美元 。 要么每个节点的需求必须由一次巡演( 无法预见 ) 或多场巡( oplisteal ) 。 通用图表的最已知的直线值算值是 $2( 2) 和 美元的直径直径( 美元) 直径1 的直径( 直径) 直径( 直径) 直径) 直径1 直径( 直径) 直径) 直径) 直径1) 直径 的O1 的结果结果结果是 。