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 <= n is the number of distinct depot-vertices). There is an Omega(log(1/4-epsilon) n) hardness of approximation known even for single vehicle capacitated Dial-a-Ride [Gortz 2006]. For uncapacitated multi-vehicle Dial-a-Ride, we show that there are instances when natural lower bounds (used in our algorithm) are (Omega) over tilde (log t) factor away from the optimum. We also consider the special class of metrics induced by graphs excluding any fixed minor (e.g., planar metrics). In this case, we obtain improved guarantees of O(log(2) n) for capacitated multi-vehicle Dial-a-Ride, and O(1) for the uncapacitated problem.
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