The Nesting Bird Box Problem is ER-complete: Sharp Hardness Results for the Hidden Set Problem

TL;DR

The paper proves that the Nesting Bird Box problem, involving placing non-visible points in a polygonal domain, is ER-complete, extending complexity results with shorter proofs using advanced tools and polygons with holes.

Contribution

It establishes ER-completeness for the Nesting Bird Box problem using novel techniques and polygonal domains with holes, simplifying previous proofs.

Findings

The problem is ER-complete.

The proof is shorter and leverages new tools.

Polygonal domains with holes are used in the proof.

Abstract

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.