Searching for a line on the plane with $n$ unit speed robots is a classic online problem that dates back to the 50's, and for which competitive ratio upper bounds are known for every $n\geq 1$. In this work we improve the best lower bound known for $n=2$ robots from 1.5993 to 3. Moreover we prove that the competitive ratio is at least $\sqrt{3}$ for $n=3$ robots, and at least $1/\cos(\pi/n)$ for $n\geq 4$ robots. Our lower bounds match the best upper bounds known for $n\geq 4$, hence resolving these cases. To the best of our knowledge, these are the first lower bounds proven for the cases $n\geq 3$ of this several decades old problem.

中文翻译：

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使用 2 个或更多机器人进行海岸线搜索的下限

用 $n$ 单位速度的机器人在飞机上搜索一条线是一个经典的在线问题，可以追溯到 50 年代，并且每个 $n\geq 1$ 的竞争比率上限都是已知的。在这项工作中，我们将 $n=2$ 机器人已知的最佳下限从 1.5993 提高到 3。此外，我们证明了 $n=3$ 机器人的竞争比率至少为 $\sqrt{3}$，并且至少$1/\cos(\pi/n)$ 用于 $n\geq 4$ 机器人。我们的下限与已知的 $n\geq 4$ 的最佳上限相匹配，因此解决了这些情况。据我们所知，这些是这个几十年前问题的 $n\geq 3$ 情况下证明的第一个下界。