Suppose that a set of mobile agents, each with a predefined maximum speed, want to patrol a fence together so as to minimize the longest time interval during which a point on the fence is left unvisited. In 2011, Czyzowicz, Gąsieniec, Kosowski and Kranakis studied this problem for the settings where the fence is an interval (a line segment) and a circle, and conjectured that the following simple strategies are always optimal: for Interval Patrolling, the simple strategy partitions the fence into subintervals, one for each agent, and lets each agent move back and forth in the assigned subinterval with its maximum speed; for Circle Patrolling, the simple strategy is to choose a number r, place the r fastest agents equidistantly around the circle, and move them at the speed of the rth agent. Surprisingly, these conjectures were then proved false: schedules were found (for some settings of maximum speeds) that slightly outperform the simple strategies.

In this paper, we are interested in the ratio between the performances of optimal schedules and simple strategies. For the two problems, we construct schedules that are 4/3 times (for Interval Patrolling) and 21/20 times (for Circle Patrolling) as good, respectively, as the simple strategies. We also propose a new variant, in which we want to patrol a single point under the constraint that each agent can only visit the point some predefined time after its previous visit. We obtain some similar ratio bounds and $\mathsf{NP}$-hardness results related to this problem.

假设一组各自具有预定义的最大速度的移动代理程序希望一起巡逻围栏，以最小化不等待围栏上某个点的最长时间间隔。在2011年，Czyzowicz，Gąsieniec，Kosowski和Kranakis在栅栏是一个间隔（一条线段）和一个圆的环境中研究了此问题，并推测以下简单策略始终是最佳的：对于间隔巡逻，简单策略分区将栅栏划分为多个子间隔，每个子代理一个，并让每个代理在指定的子间隔中以最大速度来回移动；对于“环网巡逻”，简单的策略是选择一个数字r，将r个最快的特工等距放置在该圆周围，然后以[R个代理。令人惊讶的是，这些推测随后被证明是错误的：发现了一些时间表（对于某些最大速度的设置）略胜于简单策略。

在本文中，我们对最佳计划的性能与简单策略之间的比率感兴趣。对于这两个问题，我们构建的计划分别是简单策略的4/3倍（对于间隔巡逻）和21/20倍（对于循环巡逻）。我们还提出了一个新的变体，在该变体中，我们希望在每个代理只能在其上次访问后的预定时间访问该点的约束下巡逻单个点。我们得到一些相似的比率界限$\mathsf{NP}$硬度结果与此问题有关。