# Tight Analysis of the Smartstart Algorithm for Online Dial-a-Ride on the Line

## TL;DR

This paper provides a tight competitive analysis of the Smartstart algorithm for online Dial-a-Ride on the line, improving bounds and establishing tightness for certain cases, thus advancing understanding of this fundamental online problem.

## Contribution

It offers the first tight analysis of Smartstart for online Dial-a-Ride on the line, improving the upper bound from 3.41 to 2.94 and confirming the tightness of the 2 bound for closed requests.

## Key findings

- New upper bound of 2.94 for open, non-preemptive online Dial-a-Ride on the line.
- Tightness of the 2 competitive ratio for Smartstart in the closed case.
- Remaining gap between lower bound 2.04 and upper bound 2 for open case.

## Abstract

The online Dial-a-Ride problem is a fundamental online problem in a metric space, where transportation requests appear over time and may be served in any order by a single server with unit speed. Restricted to the real line, online Dial-a-Ride captures natural problems like controlling a personal elevator. Tight results in terms of competitive ratios are known for the general setting and for online TSP on the line (where source and target of each request coincide). In contrast, online Dial-a-Ride on the line has resisted tight analysis so far, even though it is a very natural online problem. We conduct a tight competitive analysis of the Smartstart algorithm that gave the best known results for the general, metric case. In particular, our analysis yields a new upper bound of 2.94 for open, non-preemptive online Dial-a-Ride on the line, which improves the previous bound of 3.41 [Krumke'00]. The best known lower bound remains 2.04 [SODA'17]. We also show that the known upper bound of 2 [STACS'00] regarding Smartstart's competitive ratio for closed, non-preemptive online Dial-a-Ride is tight on the line.