Minimum Makespan Multi-Vehicle Dial-a-Ride

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 {(s(i), t(i))}(i-1)(m), 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 epsilon [q] has its own depot-vertex r(j) epsilon V. A feasible schedule consists of a capacitated route for each vehicle (where vehicle j originates and ends at its depot r(j)) 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 r(j) at the end of its route. We study the preemptive version of multi-vehicle Dial-a-Ride, in which 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(log(3) 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