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.

翻译：在无权重在线区间选择问题中，目标在于最大化算法接受的非冲突区间数量。在不可撤销决策的传统在线模型中，即使对于随机化算法，竞争比也存在Ω(n)的下界[Bachmann等人，2013]。在一系列允许撤销接受决策的研究中，[Faigle和Nawijn 1995]在实时模型（区间按起始时间非递减顺序到达）中给出了一个贪心的1-竞争（即最优）算法。该算法在对抗性（任意顺序）模型中的自然扩展是2k-竞争的[Borodin和Karavasilis 2023]，其中最多存在k种不同的区间长度，并且这对于所有确定性算法及无记忆随机化算法均为最优。我们在随机顺序模型中研究此问题，其中对手选择实例，但在线序列是项目的一个均匀随机排列。我们考虑在实时模型和任意顺序模型中均最优的同一算法，并证明其在随机顺序到达下的竞争比上界为2.5。我们还展示了当至少存在两种不同类型的项目时，如何利用随机顺序到达以最坏情况偏差2/3提取一个随机比特。我们利用该比特对[Fung等人，2014]的弱随机算法进行去随机化，从而得到一个针对任意权重的单长度区间选择的确定性3-竞争算法。