With the popularity of drone technologies, aerial photography has become prevalent in many daily scenarios such as environment monitoring, structure inspection, law enforcement etc. A central challenge in this domain is the efficient coverage of a target area with photographs that can entirely capture the region, while respecting constraints such as the image resolution, and limited number of pictures that can be taken. This work investigates the computational complexity of covering a simple planar polygon using squares and circles. Specifically, it shows inapproximability gaps of $1.165$ (for squares) and $1.25$ (for restricted square centers) and develops a $2.828$-optimal approximation algorithm, demonstrating that these problems are computationally intractable to approximate. The intuitions of this work can extend beyond aerial photography to broader applications such as pesticide spraying and strategic sensor placement.

翻译：随着无人机技术的普及，航拍摄影在环境监测、结构检测、执法等日常场景中已广泛应用。该领域的核心挑战在于：在满足图像分辨率、有限拍摄张数等约束条件下，高效覆盖目标区域并确保其被完整捕捉。本研究探讨了使用正方形与圆形覆盖简单平面多边形的计算复杂度问题。具体而言，研究揭示了正方形覆盖问题存在1.165倍不可逼近性间隙，受限正方形中心问题存在1.25倍不可逼近性间隙，并提出了2.828倍最优近似算法，论证了此类问题在计算上难以精确逼近。本研究的理论洞见可推广至农药喷洒、传感器战略部署等更广泛的场景。