The Finite Capacity Dial-A-Ride Problem

We give the first non-trivial approximation algorithm for the Capacitated Dial-a-Ride problem: given a collection of objects located at points in a metric space, a specified destination point for each object, and a vehicle with a capacity of at most k objects, the goal is to compute a shortest tour for the vehicle in which all objects can be delivered to their destinations while ensuring that the vehicle carries at most k objects at any point in time. The problem is known under several names, including the Stacker Crane problem and the Dial-a-Ride problem. No theoretical approximation guarantees were known for this problem other than for the cases k = 1;1 and the trivial O(k) approximation for general capacity k. We give an algorithm with approximation ratioO(pk) for special instances on a class of tree metrics called height-balanced trees. Using Bartal’s recent results on the probabilistic approximation of metric spaces by tree metrics, we obtain an approximation ratio of O(pk logn log logn) for arbitrary n point metric spaces. When the points lie on a line (line metric), we provide a 2-approximation algorithm. We also consider the Dial-a-Ride problem in another framework: when the vehicle is allowed to leave objects at intermediate locations and pick them up at a later time and deliver them. For this model, we design an approximation algorithm whose performance ratio is O(1) for tree metrics and O(logn log logn) for arbitrary metrics. We also study the ratio between the values of the optimal solutions for the two versions of the problem. We show that unlike in k-delivery TSP in which all the objects are identical, this ratio is not bounded by a constant for the Dial-a-Ride problem, and it could be as large as (k2=3). Computer Science Department, Stanford University, CA 94305. Research supported by the Pierre and Christine Lamond Fellowship and in part by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR9357849,with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. yDepartment of Computer Science, The University of Texas at Dallas, Richardson, TX 75083-0688. Research supported in part by NSF Research Initiation Award CCR-9409625.