Random-Order Interval Selection
<|reference_start|>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.<|reference_end|>
@article{borodin2024random-order,
    title={Random-Order Interval Selection},
    author={Allan Borodin, Christodoulos Karavasilis},
    journal={arXiv preprint arXiv:2407.20941},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.20941},
    primaryClass={cs.DS}
}