The Multidepot Capacitated Vehicle Routing Problem (MCVRP) is a well-known variant of the classic Capacitated Vehicle Routing Problem (CVRP), where we need to route capacitated vehicles located in multiple depots to serve customers' demand such that each vehicle must return to the depot it starts, and the total traveling distance is minimized. There are three variants of MCVRP according to the property of the demand: unit-demand, splittable and unsplittable. We study approximation algorithms for $k$-MCVRP in metric graphs where $k$ is the capacity of each vehicle, and all three versions are APX-hard for any constant $k\geq 3$. Previously, Li and Simchi-Levi proposed a $(2\alpha+1-\alpha/k)$-approximation algorithm for splittable and unit-demand $k$-MCVRP and a $(2\alpha+2-2\alpha/k)$-approximation algorithm for unsplittable $k$-MCVRP, where $\alpha=3/2-10^{-36}$ is the current best approximation ratio for metric TSP. Harks et al. further improved the ratio to 4 for the unsplittable case. We give a $(4-1/1500)$-approximation algorithm for unit-demand and splittable $k$-MCVRP, and a $(4-1/50000)$-approximation algorithm for unsplittable $k$-MCVRP. Furthermore, we give a $(3+\ln2-\max\{\Theta(1/\sqrt{k}),1/9000\})$-approximation algorithm for splittable and unit-demand $k$-MCVRP, and a $(3+\ln2-\Theta(1/\sqrt{k}))$-approximation algorithm for unsplittable $k$-MCVRP under the assumption that the capacity $k$ is a fixed constant. Our results are based on recent progress in approximating CVRP.

翻译：多仓库容量受限车辆路径问题（MCVRP）是经典容量受限车辆路径问题（CVRP）的一个重要变体，其目标是在多个仓库中调度容量受限的车辆以满足客户需求，要求每辆车必须返回其出发的仓库，并最小化总行驶距离。根据需求的属性，MCVRP分为三种变体：单位需求、可分需求与不可分需求。本文研究度量图上的$k$-MCVRP（其中$k$为每辆车容量）的近似算法，所有三种版本在任意常数$k\geq 3$下均为APX-难问题。此前，Li与Simchi-Levi提出了可分与单位需求$k$-MCVRP的$(2\alpha+1-\alpha/k)$-近似算法，以及不可分$k$-MCVRP的$(2\alpha+2-2\alpha/k)$-近似算法，其中$\alpha=3/2-10^{-36}$为当前度量TSP的最佳近似比。Harks等人进一步将不可分情况下的近似比改进至4。本文给出单位需求与可分$k$-MCVRP的$(4-1/1500)$-近似算法，以及不可分$k$-MCVRP的$(4-1/50000)$-近似算法。此外，在容量$k$为固定常数的假设下，我们给出可分与单位需求$k$-MCVRP的$(3+\ln2-\max\{\Theta(1/\sqrt{k}),1/9000\})$-近似算法，以及不可分$k$-MCVRP的$(3+\ln2-\Theta(1/\sqrt{k}))$-近似算法。本文结果基于CVRP近似研究的最新进展。