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 {d _i }, each specifying a source s _i and a destination t _i , 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(logN),O(k)}-approximation algorithm for the preemptive version of the problem. In this paper has no min {O(log$^{\rm 1/4-{\it \epsilon}}$ N),k ^1 − ε}-approximation algorithm for any ε> 0 unless all problems in NP can be solved by randomized algorithms with expected running time O(n ^{polylog n}).