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)}^^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 r_j ∈ 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_jat the end of its route. We consider 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. Approximation algorithms for the single vehicle Dial-a-Ride problem (q = 1) have been considered in [3,10] Our main results are an O(log³ n)-approximation algorithm for preemptive multi-vehicle Dial-a-Ride, and an improved O(logt)-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(log^1/4 n) hardness of approximation known [9] even for single vehicle capacitated preemptive Dial-a-Ride. We also obtain an improved constant factor approximation algorithm for the uncapacitated multi-vehicle problem on metrics induced by graphs excluding any fixed minor.