In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The problem models the optimal placement of line sensors (e.g., infrared laser beams) for isolating many types of regions in a pair-wise manner for practical purposes (e.g., guarding against intrusions). The problem is NP-hard even if we want to find the minimum number of lines to separate two sets of points in the plane. Under the umbrella problem of barrier forming with minimum number of line segments, three settings are examined: barrier forming for point sets, point sets with polygonal obstacles, polygonal sets with polygonal obstacles. We describe methods for computing the optimal solution for the first two settings with the assistance of mathematical programming, and provide a 2-OPT solution for the third. We demonstrate the effectiveness of our methods through extensive simulations.

翻译：在这项工作中,我们对形成屏障的问题进行了结构和算法研究:选择将几组相互脱节的物体分开的直线段(屏障)最低数目,将几组相互脱节的物体分开在平面上。问题模型为以双向方式将许多类型的区域隔离开来(例如,防范入侵)的线感传感器(例如,红外激光束)的最佳位置(例如,红外线激光束),在实际目的上以双向方式将许多类型的区域隔离开来(例如,防范入侵)。问题在于NP-硬性。即使我们想找到将平面两组点分开的最起码的线条,问题也是很硬的。在以最小数目线段形成屏障的伞状问题下,我们检查了三个设置:点形屏障形成屏障、多边形障碍的点、多边形障碍的多边形装置。我们描述了在数学编程的帮助下为前两个环境计算最佳解决办法的方法,并为第三个区域提供了2-OPT解决方案。我们通过广泛的模拟展示了我们的方法的有效性。