Lower Bound for Sculpture Garden Problem

TL;DR

This paper proves that in the worst case, the number of guards needed to define any polygon in the Sculpture Garden Problem is at least n-2, confirming a conjecture in art gallery problem variants.

Contribution

It provides a conclusive proof for the conjecture that n-2 guards are necessary in the worst case for the Sculpture Garden Problem.

Findings

Confirmed the conjecture that n-2 guards are needed in the worst case.

Established a lower bound for guard placement in the problem.

Contributed to the theoretical understanding of art gallery variants.

Abstract

The purpose of the current study is to investigate a special case of art gallery problem, namely Sculpture Garden Problem. In the said problem, for a given polygon $P$, the ultimate goal is to place the minimum number of guards to define the interior polygon $P$ by applying a monotone Boolean formula composed of the guards. As the findings indicate, the conjecture about the issue that in the worst case, $n-2$ guards are required to describe any $n$-gon (Eppstein et al. 2007) can be conclusively proved.