Lower Bound for Sculpture Garden Problem: Localization of IoT Devices

The purpose of the current study is to investigate a special case of art gallery problem, namely a sculpture garden problem. In this problem, for a given polygon P, the ultimate goal is to place the minimum number of guards (landmarks) to define the interior polygon P by applying a monotone Boolean formula composed of the guards. Using this problem, it can replace the operation-based method with time-consuming, pixel-based algorithms. So, the processing time of some problems in the fields of machine vision, image processing and gamification can be strongly reduced. The problem has also many applications in mobile device localization in the Internet of Things (IoT). An open problem in this regard is the proof of Eppstein's conjecture, which has remained an open problem since 2007. According to his conjecture, in the worst case, n-2 vertex guards are required to describe any n-gon. In this paper, a lower bound is introduced for the special case of this problem (natural vertex guard), which shows that if a polygon can be defined with natural vertex guards, then n-2 is a lower bound.