The seat reservation problem

We investigate the problem of giving seat reservations on-line. We assume that a train travels from a start station to an end station, stopping at k stations, including the first and last. Reservations can be made for any trip going from any station to any later station. The train has a fixed number of seats. The seat reservation system attempts to maximize income. We consider the case in which all tickets have the same price and the case in which the price of a ticket is proportional to the length of the trip. For both cases we prove upper and lower bounds of Θ(1/k) on the competitive ratio of any ‘fair’ deterministic algorithm. We also define the accommodating ratio which is similar to the competitive ratio except that the only sequences of requests allowed are sequences for which the optimal off-line algorithm could accommodate all requests. We prove upper and lower bounds of Θ(1) on the accommodating ratio of any ‘fair’ deterministic algorithm, in the case in which all tickets have the same price, but Θ(1/k) in the case in which the ticket price is proportional to the length of the trip. The most surprising of these results is that all ‘fair’ algorithms are at least 1/2-accommodating when all tickets have the same price. We prove similar results bounding the performance of any ‘fair’ randomized algorithm against an adaptive on-line adversary. We also consider concrete algorithms; more specifically, First-Fit and Best-Fit.