A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We consider searching schedules in the following one-dimensional geometric domains: the cycle of a known circumference L, the finite straight line segment of a known length L, and the semi-infinite line $[0,+\infty )$.

We first consider the online Beachcombers' Problem (i.e. the scenario when the robots do not know in advance the length of the segment to be searched), where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as $S={\inf }_{\ell }\frac{\ell }{t_A(\ell )}$, where $t_A(\ell )$ denotes the time when the search of the segment $[0,\ell ]$ is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over $\ell \in {\mathbb{N}}^{⁎}$ or $\ell \ge 1$. We prove that the LeapFrog algorithm, which was proposed in Czyzowicz et al. (2015) [12], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting.

For the offline version of the Beachcombers' Problem (i.e. the scenario when the robots know in advance the length of the segment to be searched), we consider the t-source Beachcombers' Problem (i.e. all robots start from a fixed number $t\ge 1$ of starting positions) on the cycle and on the finite segment. For the t-source Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the 2t-source Beachcombers' Problem on a finite segment. In consequence, by using results from Czyzowicz et al. (2014) [13], we prove that the 1-source Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the t-source variant of the Beachcombers' Problem on the cycle and on the finite segment, we also derive efficient approximation algorithms.

One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in Czyzowicz et al. (2014) [13], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

一组移动机器人（泳客）必须共同搜索给定域的每个点。在任何给定的时刻，每个机器人都可以处于步行模式或搜索模式。假定每个机器人的最大允许搜索速度严格小于其最大允许的行走速度。如果至少一个机器人以搜索模式访问该域，则将搜索该域的一个点。Beachcombers的问题在于为机器人制定有效的时间表（算法），以便尽可能快地集体搜索给定域的所有点。我们考虑在以下一维几何域中搜索计划：已知周长L的周期，已知长度L的有限直线段和半无限线$[0，+\infty ）$。

我们首先考虑在线比奇科默斯问题（即，机器人事先不知道要搜索的段的长度的情况），其中机器人最初位于半无限线的原点。试图设计具有最大速度S的时间表A，定义为$\mathrm{小号}={\mathrm{信息}}_{\ell }\frac{\ell }{{Ť}_{\mathrm{一种}}（\ell ）}$，在哪里 ${Ť}_{\mathrm{一种}}（\ell ）$ 表示段搜索的时间 $[0，\ell ]$在A下完成。我们根据问题是否被接管来考虑问题的离散和连续版本$\ell \in {\mathbb{ñ}}^{⁎}$ 要么 $\ell \ge \mathrm{1个}$。我们证明了在Czyzowicz等人中提出的LeapFrog算法。（2015）[12]，实际上在离散情况下是最优的。这肯定是该论文中的一个猜想。我们还将展示如何将此结果扩展到更通用的连续在线设置。

对于Beachcombers问题的离线版本（即，机器人事先知道要搜索的段的长度的情况），我们考虑t源Source Beachcombers问题（即，所有机器人都从固定数量开始$Ť\ge \mathrm{1个}$周期和有限段上的起始位置）。对于周期上的t源Beachcombers问题，我们表明最优解的结构与有限段上2 t源Beachcombers问题的最优解的结构相同。因此，通过使用Czyzowicz等人的结果。（2014）[13]，我们证明了循环上的一源Beachcombers问题是NP-hard的，并且推导了该问题的近似算法。对于循环上和有限段上的比奇科姆问题的t源变量，我们还导出了有效的近似算法。

我们工作的一项重要贡献是，在我们讨论的离线Beachcombers问题的所有变体中，我们都允许机器人在其各自起始位置的两侧更改运动方向和域的搜索点。与Czyzowicz等人的模型相比，这代表了重要的概括。（2014）[13]，其中每个机器人都有固定的运动方向，该方向被指定为问题解决方案的一部分。我们设法证明方向的改变不会帮助机器人达到最佳状态。