On The Computational Complexity of Minimum Aerial Photographs for Planar Region Coverage

TL;DR

This paper studies the computational difficulty of optimally covering planar regions with aerial photographs, establishing inapproximability bounds and providing a near-optimal algorithm, with implications for drone-based coverage tasks.

Contribution

It proves inapproximability results for covering polygons with squares and circles and introduces a 2.828-approximation algorithm, highlighting the problem's computational intractability.

Findings

Inapproximability gap of 1.165 for squares

Inapproximability gap of 1.25 for restricted square centers

A 2.828-approximate algorithm for coverage problems

Abstract

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.