**The Seat Reservation Problem***Joan Boyar, Kim S. Larsen*

Algorithmica 25(4): 403-417, 1999.

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 *Theta(1/k)* on the competitive
ratio of any ``fair''
deterministic algorithm.
We also define the {\em 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 price, but *Theta(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.

Last modified: September 21, 1999. Joan Boyar (joan@imada.sdu.dk)