In the (Nesting) Bird Box Problem we are given a polygonal domain P and a number k and we want to know if there is a set B of k points inside P such that no two points in B can see each other. The underlying idea is that each point represents a birdhouse and many birds only use a birdhouse if there is no other occupied birdhouse in its vicinity. We say two points a,b see each other if the open segment ab intersects neither the exterior of P nor any vertex of P. We show that the Nesting Bird Box problem is ER-complete. The complexity class ER can be defined by the set of problems that are polynomial time equivalent to finding a solution to the equation $p(x) = 0$, with $x\in R^n$ and $p\in $Z[X_1,...,X_n]$. The proof builds on the techniques developed in the original ER-completeness proof of the Art Gallery problem. However our proof is significantly shorter for two reasons. First, we can use recently developed tools that were not available at the time. Second, we consider polygonal domains with holes instead of simple polygons.

翻译：在（嵌套）鸟箱问题中，给定一个多边形区域P和一个整数k，我们想要判断是否存在一个包含k个点的集合B ⊆ P，使得B中任意两点彼此不可见。其基本思想是，每个点代表一个鸟屋，而许多鸟类仅在周围没有其他被占用的鸟屋时才使用该鸟屋。我们称两点a和b彼此可见，当且仅当开线段ab既不与P的外部相交，也不穿过P的任意顶点。我们证明嵌套鸟箱问题是ER完全的。复杂性类ER可定义为与求解方程$p(x) = 0$（其中$x\in R^n$, $p\in \mathbb{Z}[X_1,...,X_n]$）多项式时间等价的问题集合。该证明基于艺术画廊问题原始ER完全性证明中的技术，但我们的证明显著更短，原因有二：其一，我们能够利用当时尚未可用的最新工具；其二，我们考虑的是带孔的多边形区域而非简单多边形。