We consider the problem of online interval scheduling on a single machine, where intervals arrive online in an order chosen by an adversary, and the algorithm must output a set of non-conflicting intervals. Traditionally in scheduling theory, it is assumed that intervals arrive in order of increasing start times. We drop that assumption and allow for intervals to arrive in any possible order. We call this variant any-order interval selection (AOIS). We assume that some online acceptances can be revoked, but a feasible solution must always be maintained. For unweighted intervals and deterministic algorithms, this problem is unbounded. Under the assumption that there are at most $k$ different interval lengths, we give a simple algorithm that achieves a competitive ratio of $2k$ and show that it is optimal amongst deterministic algorithms, and a restricted class of randomized algorithms we call memoryless, contributing to an open question by Adler and Azar 2003; namely whether a randomized algorithm without access to history can achieve a constant competitive ratio. We connect our model to the problem of call control on the line, and show how the algorithms of Garay et al. 1997 can be applied to our setting, resulting in an optimal algorithm for the case of proportional weights. We also consider the case of intervals arriving in a random order, and show that for single-lengthed instances, a one-directional algorithm (i.e. replacing intervals in one direction), is the only deterministic memoryless algorithm that can possibly benefit from random arrivals. Finally, we briefly discuss the case of intervals with arbitrary weights.

翻译：我们研究单机上的在线区间调度问题，其中区间以对手选择的任意顺序在线到达，算法需输出一组无冲突的区间。传统调度理论假设区间按开始时间递增顺序到达，我们摒弃该假设，允许区间以任意可能顺序到达，并将此变体称为任意顺序区间选择（AOIS）。我们假设部分在线接受的区间可被撤销，但必须始终维护一个可行解。对于无权区间和确定性算法，该问题无界。在最多存在$k$种不同区间长度的假设下，我们给出一个达到竞争比$2k$的简单算法，并证明该算法在确定性算法及一类称为无记忆的受限随机算法中是最优的，从而回应了Adler与Azar 2003年提出的开放问题：即无历史访问权限的随机算法能否实现常数竞争比。我们将模型与线上呼叫控制问题建立联系，展示如何将Garay等人1997年的算法应用于我们的场景，从而在比例权重情形下得到最优算法。此外，我们考虑区间随机顺序到达的情形，证明对于单长度实例，单向算法（即仅沿一个方向替换区间）是唯一可能从随机到达中获益的确定性无记忆算法。最后，我们简要讨论任意权重区间的情形。