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 Bundes-bahn (the German railway company) which entitles its holder to a 50% price reduction on nearly all train tickets. It costs 240 DM, and it is valid for 12 months. 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 where to she will travel next. We show that the greedy algorithm applied by most travelers and clerks at ticket ooces 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 random-ized 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. It turns out that the optimal competitive ratio only depends on the price reduction factor (50% for the German Bahncard Problem), but does not depend on the price or validity period of a Bahncard.