Algorithms for the On-Line Travelling Salesman1

Abstract. In this paper the problem of efficiently serving a sequence of requests presented in an on-line fashion located at points of a metric space is considered. We call this problem the On-Line Travelling Salesman Problem (OLTSP). It has a variety of relevant applications in logistics and robotics. We consider two versions of the problem. In the first one the server is not required to return to the departure point after all presented requests have been served. For this problem we derive a lower bound on the competitive ratio of 2 on the real line. Besides, a 2.5 -competitive algorithm for a wide class of metric spaces, and a 7/3 -competitive algorithm for the real line are provided. For the other version of the problem, in which returning to the departure point is required, we present an optimal 2 -competitive algorithm for the above-mentioned general class of metric spaces. If in this case the metric space is the real line we present a 1.75 -competitive algorithm that compares with a \approx 1.64 lower bound.

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

[2]  Toshihide Ibaraki,et al.  Vehicle Scheduling on a Tree with Release and Handling Times , 1993, ISAAC.

[3]  Robert E. Tarjan,et al.  Self-adjusting binary search trees , 1985, JACM.

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

[5]  Allan Borodin,et al.  On the power of randomization in online algorithms , 1990, STOC '90.

[6]  Marius M. Solomon,et al.  Routing and scheduling on a shoreline with release times , 1990 .

[7]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .

[8]  David S. Johnson,et al.  Near-optimal bin packing algorithms , 1973 .

[9]  Ran El-Yaniv,et al.  Competitive analysis of financial games , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[10]  Marius M. Solomon,et al.  Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints , 1987, Oper. Res..

[11]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

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

[13]  Lyle A. McGeoch,et al.  Competitive Algorithms for Server Problems , 1990, J. Algorithms.

[14]  Jacques Desrosiers,et al.  Time Window Constrained Routing and Scheduling Problems: a Survey , 1987 .

[15]  Bala Kalyanasundaram,et al.  Constructing Competitive Tours from Local Information , 1993, Theor. Comput. Sci..

[16]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[17]  Chul E. Kim,et al.  Approximation Algorithms for Some Routing Problems , 1978, SIAM J. Comput..

[18]  Robert E. Tarjan,et al.  Amortized efficiency of list update and paging rules , 1985, CACM.

[19]  Greg N. Frederickson.  A Note on the Complexity of a Simple Transportation Problem , 1993, SIAM J. Comput..

[20]  Eric Torng,et al.  The k-client problem , 1997, SODA '97.