The multi-vehicle Dial-a-Ride problem (mDaR) involves...

Dial-a-Ride problems consist of a set V of n vertices in a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs {(si, ti)}mi=1, where each object requires to be moved from its source to destination vertex. In the multi-vehicle Dial-a-Ride problem, there are q vehicles each having capacity k and where each vehicle j ∈ [q] has its own depot-vertex rj ∈ V. A feasible schedule consists of a capacitated route for each vehicle (where vehicle j originates and ends at its depot rj) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e. makespan) of vehicles, where the completion time of vehicle j is the time when it returns to its depot rj at the end of its route. We study the preemptive version of multi-vehicle Dial-a-Ride, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log3 n)-approximation algorithm for preemptive multi-vehicle Dial-a-Ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint (here t ≤ n is the number of distinct depot-vertices). There is an Ω(log1/4− n) hardness of approximation known even for single vehicle capacitated Dial-a-Ride [Gørtz 2006]. For uncapacitated multi-vehicle Dial-a-Ride, we show that there are instances where natural lower bounds (used in our algorithm) are Ω̃(log t) factor away from the optimum