We study the online maximum coverage problem on a line, in which, given an online sequence of sub-intervals (which may intersect among each other) of a target large interval and an integer $k$, we aim to select at most $k$ of the sub-intervals such that the total covered length of the target interval is maximized. The decision to accept or reject each sub-interval is made immediately and irrevocably (no preemption) right at the release timestamp of the sub-interval. We comprehensively study different settings of this problem regarding both the length of a released sub-interval and the total number of released sub-intervals. We first present lower bounds on the competitive ratio for the settings concerned in this paper, respectively. For the offline problem where the sequence of all the released sub-intervals is known in advance to the decision-maker, we propose a dynamic-programming-based optimal approach as the benchmark. For the online problem, we first propose a single-threshold-based deterministic algorithm SOA by adding a sub-interval if the added length exceeds a certain threshold, achieving competitive ratios close to the lower bounds, respectively. Then, we extend to a double-thresholds-based algorithm DOA, by using the first threshold for exploration and the second threshold (larger than the first one) for exploitation. With the two thresholds solved by our proposed program, we show that DOA improves SOA in the worst-case performance. Moreover, we prove that a deterministic algorithm that accepts sub-intervals by multi non-increasing thresholds cannot outperform even SOA.

中文翻译：

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在线最大$ k $时间间隔覆盖问题

我们在线研究在线最大覆盖率问题，其中，给定目标大间隔和整数$ k $的子间隔的在线序列（可能彼此相交），我们旨在选择最多$ k $个子间隔，以使目标间隔的总覆盖长度最大化。在该子时间间隔的发布时间戳记上，立即且不可撤消地（无抢占）地决定接受还是拒绝每个子时间间隔。我们就释放子间隔的长度和释放子间隔的总数，全面研究了此问题的不同设置。我们首先针对本文所涉及的环境分别提出竞争比率的下限。对于决策者事先知道所有已释放子间隔的顺序的离线问题，我们提出了一种基于动态编程的最佳方法作为基准。对于在线问题，我们首先提出一种基于单阈值的确定性算法SOA，方法是在添加的长度超过一定阈值时添加子间隔，从而分别实现接近下限的竞争率。然后，通过使用用于探索的第一个阈值和用于利用的第二个阈值（大于第一个阈值），扩展到基于双阈值的算法DOA。通过我们提出的程序解决的两个阈值，我们证明了DOA在最坏的情况下可以提高SOA。此外，