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

TL;DR

This paper provides a tighter analysis of the Lazy algorithm for the open online dial-a-ride problem, improving the upper bounds on its competitive ratio in general metric spaces and on the half-line.

Contribution

The paper offers the first tight analysis of the Lazy algorithm, establishing improved competitive ratios that beat previous lower bounds for schedule-based algorithms.

Findings

Competitive ratio 2.457 on general metric spaces

Competitive ratio 2.366 on the half-line

First upper bound to beat known lower bounds of 2.5

Abstract

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.