The multi-vehicle dial-a-ride problem (mDaRP) is a fundamental vehicle routing problem with pickups and deliveries, widely applicable in ride-sharing, economics, and transportation. Given a set of $n$ locations, $h$ vehicles of identical capacity $λ$ located at various depots, and $m$ ride requests each defined by a source and a destination, the goal is to plan non-preemptive routes that serve all requests while minimizing the total travel distance, ensuring that no vehicle carries more than $λ$ passengers at any time. The best-known approximation ratio for the mDaRP remains $\mathcal{O}(\sqrtλ\log m)$. We propose two simple algorithms: the first achieves the same approximation ratio of $\mathcal{O}(\sqrtλ\log m)$ with improved running time, and the second attains an approximation ratio of $\mathcal{O}(\sqrt{\frac{m}λ})$. A combination of them yields an approximation ratio of $\mathcal{O}(\sqrt[4]{n}\log^{\frac{1}{2}}n)$ under $m=Θ(n)$. Moreover, for the case $m\gg n$, by extending our algorithms, we derive an $\mathcal{O}(\sqrt{n\log n})$-approximation algorithm, which also improves the current best-known approximation ratio of $\mathcal{O}(\sqrt{n}\log^2n)$ for the classic (single-vehicle) DaRP, obtained by Gupta et al. (ACM Trans. Algorithms, 2010).

翻译：多车辆即时配送问题（mDaRP）是车辆路径规划中具有接送需求的基础性问题，在拼车服务、经济学和交通运输领域具有广泛应用。给定包含 $n$ 个地点的集合、位于不同站点的 $h$ 辆容量均为 $λ$ 的相同车辆，以及 $m$ 个由起点和终点定义的出行请求，目标是在确保任何时刻每辆车载客数不超过 $λ$ 的前提下，规划服务所有请求的非抢占式行驶路线，以最小化总行驶距离。目前 mDaRP 最著名的近似比为 $\mathcal{O}(\sqrtλ\log m)$。我们提出了两种简单算法：第一种在保持 $\mathcal{O}(\sqrtλ\log m)$ 近似比的同时提升了运行时间；第二种实现了 $\mathcal{O}(\sqrt{\frac{m}λ})$ 的近似比。在 $m=Θ(n)$ 条件下，结合两种算法可获得 $\mathcal{O}(\sqrt[4]{n}\log^{\frac{1}{2}}n)$ 的近似比。此外，针对 $m\gg n$ 的情况，通过扩展我们的算法，我们推导出 $\mathcal{O}(\sqrt{n\log n})$ 近似算法，该结果也改进了 Gupta 等人（ACM Trans. Algorithms, 2010）针对经典（单车辆）DaRP 问题提出的当前最佳近似比 $\mathcal{O}(\sqrt{n}\log^2n)$。