On-Line Maximizing the Number of Items Packed in Variable-Sized Bins
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
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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. Passengers may attempt to make reservations at any point before they get on the train. Reservations can be made 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 cost the same amount and the case in which the cost of a ticket is proportional to the length of the trip. In these cases, we prove upper and lower bounds of $\Theta(1/k)$ 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 $\Theta(1)$ on the accommodating ratio of any ``fair'''' deterministic algorithm, in the case in which all tickets have the same cost, but $\Theta(1/k)$ in the case in which the ticket cost 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 cost. We also consider concrete algorithms; more specifically, First-Fit and Best-Fit.