Hardness of preemptive finite capacity dial-a-ride

Abstract

In the Finite Capacity Dial-a-Ride problem the input is a metric space, a set of objects {di}, each specifying a source Si and a destination ti, 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-ε N), k1-ε-approximation algorithm for any ε > 0 unless all problems in NP can be solved by randomized algorithms with expected running time O(npolylogn). © Springer-Verlag Berlin Heidelberg 2006.