In the open online dial-a-ride problem, a single server has to deliver transportation requests appearing over time in some metric space, subject to minimizing the completion time. We improve on the best known upper bounds on the competitive ratio on general metric spaces and on the half-line, for both the preemptive and non-preemptive version of the problem. We achieve this by revisiting the algorithm $\textsc{Lazy}$ recently suggested in [WAOA, 2022] and giving an improved and tight analysis. More precisely, we show that it has competitive ratio $2.457$ on general metric spaces and $2.366$ on the half-line. This is the first upper bound that beats known lower bounds of 2.5 for schedule-based algorithms as well as the natural $\textsc{Replan}$ algorithm.

翻译：在公开在线拨叫运输问题中，单个服务器需在度量空间中随时间处理运输请求，目标是最小化完成时间。我们改进了该问题在抢占式与非抢占式版本下，针对一般度量空间与半直线的竞争比已知上界。这一改进通过重新审视近期[WAOA, 2022]提出的$\textsc{Lazy}$算法并给出更优的紧致分析实现。具体而言，我们证明该算法在一般度量空间上的竞争比为$2.457$，在半直线上为$2.366$。这是首个突破基于调度算法已知下界2.5以及自然$\textsc{Replan}$算法的上界结果。