On-line single-server dial-a-ride problems

In this paper results on the dial-a-ride problem with a single server are presented. Requests for rides consist of two points in a metric space, a source and a destination. A ride has to be made by the server from the source to the destination. The server travels at unit speed in the metric space and the objective is to minimize some function of the delivery times at the destinations. We study this problem in the natural on-line setting. Calls for rides come in while the server is traveling. This models e.g. the taxi problem, or, if the server has capacity more than 1 a minibus or courier service problem. For the version of this problem in which the server has infinite capacity having as objective minimization of the time the last destination is served, we design an algorithm that has competitive ratio 2. We also show that this is best possible, since no algorithm can have competitive ratio better than 2 independently of the capacity of the servers. Besides, we give a simple 2.5-competitive algorithm for the case with finite capacity. Then we study the on-line problem with objective minimization of the sum of completion times of the rides. We prove a lower bound on the competitive ratio of any algorithm of 1 + \sqrt{2} for a server with any capacity and of 3 for a server with capacity 1. Finally, we present the first competitive algorithm for the case the server has infinite capacity and the metric space is the real line. The algorithm has competitive ratio 15. Keywords: Dial-a-ride, on-line optimization, competitive analysis.

[1]  Ricardo A. Baeza-Yates,et al.  Searching in the Plane , 1993, Inf. Comput..

[2]  David P. Williamson,et al.  Scheduling parallel machines on-line , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[3]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[4]  Greg N. Frederickson,et al.  Nonpreemptive Ensemble Motion Planning on a Tree , 1993, J. Algorithms.

[5]  Martin Grötschel,et al.  Combinatorial Online Optimization in Practice , 1998 .

[6]  Martin W. P. Savelsbergh,et al.  The General Pickup and Delivery Problem , 1995, Transp. Sci..

[7]  Madhu Sudan,et al.  The minimum latency problem , 1994, STOC '94.

[8]  Mikhail J. Atallah,et al.  Efficient Solutions to Some Transportation Problems with Applications to Minimizing Robot Arm Travel , 1988, SIAM J. Comput..

[9]  Chul E. Kim,et al.  Approximation algorithms for some routing problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[10]  Apa Arjen Vestjens.  On-line machine scheduling , 1997 .

[11]  Greg N. Frederickson,et al.  Preemptive Ensemble Motion Planning on a Tree , 1992, SIAM J. Comput..

[12]  Leen Stougie,et al.  Algorithms for the On-Line Travelling Salesman1 , 2001, Algorithmica.

[13]  L. Bodin.  ROUTING AND SCHEDULING OF VEHICLES AND CREWS–THE STATE OF THE ART , 1983 .

[14]  Cynthia A. Phillips,et al.  Scheduling Jobs that Arrive Over Time (Extended Abstract) , 1995, WADS.

[15]  Moses Charikar,et al.  The finite capacity dial-a-ride problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  Dih Jiun Guan,et al.  Routing a Vehicle of Capacity Greater than one , 1998, Discret. Appl. Math..