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）鸟盒问题中，我们给定一个多边形区域P和一个数字k，需要判断是否存在一个由P内k个点构成的集合B，使得B中任意两点彼此不可见。其核心思想是每个点代表一个鸟屋，且许多鸟类仅在附近没有其他被占据鸟屋时才会使用该鸟屋。我们称两点a、b彼此可见，若开线段ab既不与P的外部相交，也不与P的任何顶点相交。我们证明嵌套鸟盒问题是ER完全的。复杂性类ER可定义为与求解方程$p(x)=0$（其中$x\in R^n$且$p\in \mathbb{Z}[X_1,...,X_n]$）在多项式时间等价的决策问题集合。该证明基于美术馆问题原始ER完全性证明中发展的技术。然而我们的证明显著更短，原因有二：其一，我们能够使用当时尚不存在的近期开发工具；其二，我们考虑带孔洞的多边形区域而非简单多边形。