Let R be a connected and closed region in the plane and let S be a set of n points (representing mobile sensors) in the interior of R. We think of R's boundary as a barrier which needs to be monitored. This gives rise to the barrier coverage problem, where one needs to move the sensors to the boundary of R, so that every point on the boundary is covered by one of the sensors. We focus on the variant of the problem where the goal is to place the sensors on R's boundary, such that the distance (along R's boundary) between any two adjacent sensors is equal to R's perimeter over n and the sum of the distances traveled by the sensors is minimum. In this paper, we consider the cases where R is either a circle or a convex polygon. We present a PTAS for the circle case and explain how to overcome the main difficulties that arise when trying to adapt it to the convex polygon case. Our PTASs are significantly faster than the previous ones due to Bhattacharya et al. [4]. Moreover, our PTASs require efficient solutions to problems, which, as we observe, are equivalent to the circle-restricted and line-restricted Weber problems. Thus, we also devise efficient PTASs for these Weber problems.

设R为平面中的连通区域和闭合区域，设S为R内部的一组n点（表示移动传感器）。我们认为R的边界是需要监视的障碍。这引起了障碍物覆盖问题，其中需要将传感器移动到R的边界，从而边界上的每个点都被其中一个传感器覆盖。我们关注问题的变体，目标是将传感器放置在R的边界上，以使任意两个相邻传感器之间的距离（沿着R的边界）等于R在n上的周长并且传感器行进的距离之和最小。在本文中，我们考虑R为圆形或凸多边形的情况。我们为圆盒提供了一个PTAS，并解释了如何克服尝试使其适应凸多边形盒时出现的主要困难。由于Bhattacharya等人的研究，我们的PTAS比以前的PTAS显着更快。[4]。此外，我们的PTAS需要有效的解决方案，正如我们所观察到的，这些解决方案等效于圆受限和行受限的韦伯问题。因此，我们还针对这些Weber问题设计了有效的PTAS。