Improved Approximations for Dial-a-Ride Problems

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).

💡 Research Summary

The paper tackles the multi‑vehicle dial‑a‑ride problem (mDaRP), a fundamental routing task where h identical vehicles of capacity λ must serve m pickup‑delivery requests on a metric graph while minimizing total travel distance. The best known approximation before this work was O(√λ log m) with a high‑order running time. The authors present two conceptually simple algorithms that improve both the approximation quality and the computational efficiency.

Algorithm 1 adapts the iterated tour partition (ITP) technique from the capacitated vehicle routing problem. It builds two metric subgraphs: one on all sources plus vehicle depots, and another on all destinations plus depots. Approximate TSP tours are computed on each subgraph, and the tours are forced to be consistent (i.e., they visit the same request indices in the same order). The tours are then split into fragments respecting the vehicle capacity, each fragment is doubled, a minimum‑weight edge linking the source fragment to its matching destination fragment is added, and metric shortcutting yields feasible routes. This yields the same O(√λ log m) approximation as the prior state‑of‑the‑art but runs in O(m²) time, a dramatic speed‑up over the previous O(m³ log λ + m² λ² log λ) algorithm.

Algorithm 2 relies on a 2‑approximation for the multi‑depot TSP (mTSP). The authors construct a bipartite complete graph where one side represents vehicles (depots) and the other side represents requests. Solving mTSP on this graph produces two consistent tour sets for sources and destinations. They then apply a λ‑Steiner forest decomposition, a technique introduced by Gupta et al., to partition the requests into groups that each fit into a vehicle of capacity λ. This yields an approximation ratio of O(√(m/λ)) with a running time of O(m² log m).

By selecting the better of the two algorithms for a given instance, the combined guarantee becomes O(min{√λ log m, √(m/λ)}). When m = Θ(n) (the typical dense‑request regime), this simplifies to O(n¹⁄⁴ log¹⁄² n), breaking the previous O(√n) barrier for m ≪ n². Moreover, for the case m ≫ n, the authors extend their methods with the λ‑Steiner forest approach to obtain an O(√(n log n)) approximation. This improves the long‑standing bound for the classic single‑vehicle DaRP from O(√n log² n) to O(√n log n).

The paper also introduces new structural insights: representing each request as a single vertex in a metric graph enables the use of CVRP techniques, and the existence of consistent source/destination tours is proved via the mTSP construction. These insights unify concepts from CVRP, TSP, and Steiner forest problems, leading to algorithms that are both theoretically stronger and practically more efficient. Overall, the work advances the state of the art for both multi‑vehicle and single‑vehicle dial‑a‑ride problems, offering simpler, faster, and tighter approximation algorithms.