Hardness of Preemptive Finite Capacity Dial-a-Ride

In the Finite Capacity Dial-a-Ride problem the input is a metric space, a set of objects, each specifying a source and a destination, and an integer k---the capacity of the vehicle used for making the deliveries. The goal is to compute a shortest tour for the vehicle in which all objects can be delivered from their sources to their destinations while ensuring that the vehicle carries at most k objects at any point in time. In the preemptive version an object may be dropped at intermediate locations and picked up later and delivered. Let N be the number of nodes in the input graph. Charikar and Raghavachari [FOCS '98] gave a min{O(log N),O(k)}-approximation algorithm for the preemptive version of the problem. In this paper we show that the preemptive Finite Capacity Dial-a-Ride problem has no $min{O(log^{1/4-\epsilon}N),k^{1-\epsilon}}$-approximation algorithm for any $\epsilon>0$ unless all problems in NP can be solved by randomized algorithms with expected running time $O(n^{loglog n})$.