Improved Approximations for Dial-a-Ride Problems
Jingyang Zhao, Mingyu Xiao
TL;DR
This work advances approximation algorithms for the non-preemptive multi-vehicle Dial-a-Ride Problem (mDaRP) on metric graphs. It introduces two simple, practical algorithms: Alg.1 with an $\mathcal{O}(\sqrt{\lambda}\log m)$-approximation and $\mathcal{O}(m^2)$ time, and Alg.2 with an $\mathcal{O}(\sqrt{m/\lambda})$-approximation and $\mathcal{O}(m^2\log m)$ time, then combines them to achieve $\mathcal{O}(n^{1/4}\log^{1/2} n)$ when $m=\Theta(n)$ and extends to $m \gg n$ to yield $\mathcal{O}(\sqrt{n\log n})$-approximation, improving the best-known bounds for both multi-vehicle and single-vehicle DaRP. The methods rely on a CVRP-inspired two-tour reduction via a minimum-weight mTSP on a gadget graph and a Steiner-forest decomposition to create efficient, per-vehicle walks, underpinned by new structural properties. An enhanced algorithmic framework combines these ideas to attain the $\mathcal{O}(\sqrt{n\log n})$ bound in the regime $m=\Omega(n^2)$, offering practical, scalable solutions for large-scale mDaRP instances and pushing beyond long-standing barriers in DaRP approximations.
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 $λ$ 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 $λ$ passengers at any time. The best-known approximation ratio for the mDaRP remains $\mathcal{O}(\sqrtλ\log m)$. We propose two simple algorithms: the first achieves the same approximation ratio of $\mathcal{O}(\sqrtλ\log m)$ with improved running time, and the second attains an approximation ratio of $\mathcal{O}(\sqrt{\frac{m}λ})$. A combination of them yields an approximation ratio of $\mathcal{O}(\sqrt[4]{n}\log^{\frac{1}{2}}n)$ under $m=Θ(n)$. 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).