On the Bahncard Problem
In this paper, we generalize the Ski-Rental Problem to the Bahncard Problem which is an online problem of practical relevance for all travelers. The Bahncard is a railway pass of the Deutsche Bundesbahn (the German railway company) which entitles its holder to a 50% price reduction on nearly all train tickets. It costs 240DM, and it is valid for 12 months. Similar bus or railway passes can be found in many other countries.
For the common traveler, the decision at which time to buy a Bahncard is a typical online problem, because she usually does not know when and to which place she will travel next. We show that the greedy algorithm applied by most travelers and clerks at ticket offices is not better in the worst case than the trivial algorithm which never buys a Bahncard. We present two optimal deterministic online algorithms, an optimistic one and and a pessimistic one. We further give a lower bound for randomized online algorithms and present an algorithm which we conjecture to be optimal; a proof of the conjecture is given for a special case of the problem.
KeywordsCompetitive Ratio Online Algorithm Springer Lecture Note Ticket Price Request Sequence
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- 1.S. Albers and H. Koga. New on-line algorithms for the page replication problem. In Proceedings of the 4th Scandinavian Workshop on Algorithm Theory (SWAT’94). Springer Lecture Notes in Computer Science 824, pages 25–36, 1994.Google Scholar
- 2.Y. Azar, Y. Bartal, E. Feuerstein, A. Fiat, S. Leonardi, and A. Rosén. On capital investment. In Proceedings of the 23rd International Colloquium on Automata, Languages and Programming (ICALP’96). Springer Lecture Notes in Computer Science 1099, pages 429–441, 1996.Google Scholar
- 4.D.L. Black and D.D. Sleator. Competitive algorithms for replication and migration problems. Technical Report CMU-CS-89-201, Carnegie Mellon University, 1989.Google Scholar
- 5.R. Fleischer. On the Bahncard problem. Technical Report MPI-I-97-1-018, Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany, September 1997.Google Scholar
- 8.C. Lund, N. Reingold, J. Westbrook, and D. Yan. On-line distributed data management. In Proceedings of the 2nd European Symposium on Algorithms (ESA’94). Springer Lecture Notes in Computer Science 855, pages 202–214, 1994.Google Scholar
- 9.P. Raghavan. Lecture notes on randomized algorithms. Technical Report RC 15340 1/9/90, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York, 1990.Google Scholar
- 12.R.E. Tarjan. Data Structures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, Pennsylvania, 1983.Google Scholar