Improved Approximations for Dial-a-Ride Problems

TL;DR

This paper introduces two new algorithms for the multi-vehicle dial-a-ride problem, improving approximation ratios and running times, with implications for both multi-vehicle and single-vehicle scenarios in vehicle routing.

Contribution

The paper presents two simple algorithms that match or improve existing approximation ratios for the mDaRP, with faster running times and extensions to single-vehicle cases.

Findings

First algorithm achieves $ ilde{O}(rac{1}{2} ext{ approximation ratio with improved runtime.

Second algorithm attains an $O(rac{m}{ ext{lambda}})$ approximation ratio.

Extended algorithms improve approximation ratios for large request sets and single-vehicle cases.

Abstract

The multi-vehicle dial-a-ride problem (mDaRP) is a fundamental vehicle routing problem with pickups and deliveries, widely applicable in ride-sharing, economics, and transportation. Given a set of $n$ locations, $h$ vehicles of identical capacity $\lambda$ located at various depots, and $m$ ride requests each defined by a source and a destination, the goal is to plan non-preemptive routes that serve all requests while minimizing the total travel distance, ensuring that no vehicle carries more than $\lambda$ passengers at any time. The best-known approximation ratio for the mDaRP remains $\mathcal{O}(\sqrt{\lambda}\log m)$. We propose two simple algorithms: the first achieves the same approximation ratio of $\mathcal{O}(\sqrt{\lambda}\log m)$ with improved running time, and the second attains an approximation ratio of $\mathcal{O}(\sqrt{\frac{m}{\lambda}})$. A combination of them yields an approximation ratio of $\mathcal{O}(\sqrt[4]{n}\log^{\frac{1}{2}}n)$ under $m=\Theta(n)$. Moreover, for the case $m\gg n$, by extending our algorithms, we derive an $\mathcal{O}(\sqrt{n\log n})$-approximation algorithm, which also improves the current best-known approximation ratio of $\mathcal{O}(\sqrt{n}\log^2n)$ for the classic (single-vehicle) DaRP, obtained by Gupta et al. (ACM Trans. Algorithms, 2010).