Random-Order Interval Selection

In the problem of online unweighted interval selection, the objective is to maximize the number of non-conflicting intervals accepted by the algorithm. In the conventional online model of irrevocable decisions, there is an Omega(n) lower bound on the competitive ratio, even for randomized algorithms Bachmann et al. 2013. In a line of work that allows for revocable acceptances, Faigle and Nawijn 1995 gave a greedy 1-competitive (i.e. optimal) algorithm in the real-time model, where intervals arrive in order of non-decreasing starting times. The natural extension of their algorithm in the adversarial (any-order) model is 2k-competitive Borodin and Karavasilis 2023, when there are at most k different interval lengths, and that is optimal for all deterministic, and memoryless randomized algorithms. We study this problem in the random-order model, where the adversary chooses the instance, but the online sequence is a uniformly random permutation of the items. We consider the same algorithm that is optimal in the cases of the real-time and any-order models, and give an upper bound of 2.5 on the competitive ratio under random-order arrivals. We also show how to utilize random-order arrivals to extract a random bit with a worst case bias of 2/3, when there are at least two distinct item types. We use this bit to derandomize the barely random algorithm of Fung et al. 2014 and get a deterministic 3-competitive algorithm for single-length interval selection with arbitrary weights.