The Multiple-Depot Split Delivery Vehicle Routing Problem (MD-SDVRP) is a challenging problem with broad applications in logistics. The goal is to serve customers' demand using a fleet of capacitated vehicles located in multiple depots, where each customer's demand can be served by more than one vehicle, while minimizing the total travel cost of all vehicles. We study approximation algorithms for this problem. Previously, the only known result was a $6$-approximation algorithm for a constant number of depots (INFORMS J. Comput. 2023), and whether this ratio could be improved was left as an open question. In this paper, we resolve it by proposing a $(6-2\cdot 10^{-36})$-approximation algorithm for this setting. Moreover, we develop constant-factor approximation algorithms that work beyond a constant number of depots, improved parameterized approximation algorithms related to the vehicle capacity and the number of depots, as well as bi-factor approximation algorithms.

翻译：多仓库可拆分配送车辆路径问题（MD-SDVRP）是物流领域一个具有广泛应用前景的挑战性难题。该问题的目标在于利用分布于多个仓库的容量受限车辆车队满足客户需求——其中每位客户的需求可由多辆车辆共同服务——同时最小化所有车辆的总行驶成本。本文针对该问题研究近似算法。此前已知的唯一结果是针对固定仓库数量的6-近似算法（INFORMS J. Comput. 2023），而该近似比能否改进则被列为开放性问题。本文通过提出适用于该场景的$(6-2\cdot 10^{-36})$-近似算法解决了该问题。此外，我们开发了适用于非固定仓库数量的常数因子近似算法、改进的关于车辆容量与仓库数量的参数化近似算法，以及双因子近似算法。