Interval Selection in the Streaming Model
A set of intervals is independent when the intervals are pairwise disjoint. In the interval selection problem, we are given a set \(\mathbb I\) of intervals and we want to find an independent subset of intervals of largest cardinality, denoted \(\alpha (\mathbb I)\). We discuss the estimation of \(\alpha (\mathbb I)\) in the streaming model, where we only have one-time, sequential access to \(\mathbb I\), the endpoints of the intervals lie in \(\{ 1,\dots ,n \}\), and the amount of the memory is constrained.
In this paper, we study the asymptotical optimal solutions for the general airline revenue management problem, considering both customer cancelations and no-shows. We study a simple case where there is only one fare-class; the management controls the opening and closing of the fare-class, with the objective of maximizing the total revenue. We first develop a fluid control problem, where the customer arrival and cancelation processes are treated as deterministic, whereas the no. of no-shows is random. The optimal fluid solution is shown to have a simple structure of accepting up to a single time point t* and rejecting afterwards. We show that this policy acts as an upper bound for the MDP problem. Applying the optimal fluid solution to the stochastic environment, we show that this policy is asymptotically optimal on the fluid scaling.