# Improved Bounds for Open Online Dial-a-Ride on the Line

## TL;DR

This paper establishes a new lower bound of 2.0585 and improves the upper bound to 2.6662 for the competitive ratio of the open online Dial-a-Ride problem on the line, advancing theoretical understanding.

## Contribution

It provides the first strict separation between online Dial-a-Ride and TSP on the line and tightens the bounds for the problem's competitive ratio.

## Key findings

- Lower bound of 2.0585 on competitive ratio
- Improved upper bound to 2.6662
- Analysis of the algorithm is tight

## Abstract

We consider the open, non-preemptive online Dial-a-Ride problem on the real line, where transportation requests appear over time and need to be served by a single server. We give a lower bound of 2.0585 on the competitive ratio, which is the first bound that strictly separates online Dial-a-Ride on the line from online TSP on the line in terms of competitive analysis, and is the best currently known lower bound even for general metric spaces. On the other hand, we present an algorithm that improves the best known upper bound from 2.9377 to 2.6662. The analysis of our algorithm is tight.