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 discuss the case of intervals with arbitrary weights, and show how to convert the single-length algorithm of Fung et al. 2014 into a classify and randomly select algorithm that achieves a competitive ratio of 2k. Finally, we 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：我们研究单台机器上的在线区间调度问题，其中区间按照对手选择的任意顺序在线到达，算法需输出一组无冲突区间。传统调度理论假设区间按开始时间递增顺序到达，我们摒弃这一假设，允许区间以任意顺序到达，并将此变体称为任意顺序区间选择（AOIS）。我们允许撤销部分在线接受决策，但需始终维护可行解。对于无权区间和确定性算法，该问题无界。在假设区间长度至多存在$k$种不同的情况下，我们给出一个简单算法，其竞争比为$2k$，并证明该算法在确定性算法以及一类受限的无记忆随机算法中达到最优，这回应了Adler与Azar（2003）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。我们将模型与线路上的呼叫控制问题建立关联，并展示Garay等人（1997）的算法如何适用于本场景，从而在比例权重情形下获得最优算法。此外，我们讨论任意权重区间的情形，并说明如何将Fung等人（2014）的单长度区间算法转化为分类随机选择算法，实现$2k$的竞争比。最后，我们考虑区间按随机顺序到达的情形，证明对于单长度实例，单向算法（即仅沿单一方向替换区间）是唯一可能从随机到达中获益的确定性无记忆算法。我们亦简要讨论了任意权重区间的情形。