Better Bounds on the Accommodating Ratio for the Seat Reservation Problem

In a recent paper [J. Boyar and K. S. Larsen, “The seat reservation problem”, Algorithmica 25, 403-417 (1999; Zbl 0937.68156)], the seat reservation problem was investigated. It was shown that for the unit price problem, where all tickets have the same price, all “fair” algorithms are at least 1/2-accommodating, while no fair algorithm is more than (4/5+O(1/k))-accommodating, where k is the number of stations the train travels. In this paper, we design a more dextrous adversary argument, such that we improve the upper bound on the accommodating ratio to (7/9+O(1/k)), even for fair randomized algorithms against oblivious adversaries. For deterministic algorithms, the upper bound is lowered to approximately .7699. It is shown that better upper bounds exist for the special cases with n=2,3, and 4 seats. A concrete on-line deterministic algorithm First-Fit and an on-line randomized algorithm Random are also examined for the special case n=2, where they are shown to be asymptotically optimal.