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

Tight analysis of the lazy algorithm for open online dial-a-ride