Tight bounds on the competitive ratio on accomodating sequences for the seat reservation problem

The unit price seat reservation problem is investigated. The seat reservation problem is the problem of assigning seat numbers on-line to requests for reservations in a train traveling through $k$ stations. We are considering the version where all tickets have the same price and where requests are treated fairly, i.e., a request which can be fulfilled must be granted. For fair deterministic algorithms, we provide an asymptotically matching upper bound to the existing lower bound which states that all fair algorithms for this problem are $frac{1{2$-competitive on accommodating sequences, when there are at least three seats. Additionally, we give an asymptotic upper bound of $frac{7{9$ for fair randomized algorithms against oblivious adversaries. We also examine concrete on-line algorithms, First-Fit and Random, for the special case of two seats. Tight analyses of their performance are given.