Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

TL;DR

This paper studies a generalized art gallery problem involving guards with limited wall penetration, proving NP-hardness for minimum guard placement and providing exact bounds for various polygon types.

Contribution

It introduces the concept of k-transmitters as guards, proves NP-hardness for certain cases, and establishes necessary and sufficient conditions for edge 2-transmitters in different polygons.

Findings

NP-hardness of computing minimum point and edge 2-transmitters in polygons

Extension of point 2-transmitter results to orthogonal polygons

Necessary and sufficient conditions for edge 2-transmitters in various polygon classes

Abstract

We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called $k$-transmitters, model a wireless device with a signal that can pass through at most $k$ walls. We show it is NP-hard to compute a minimum cover of point 2-transmitters, point $k$-transmitters, and edge 2-transmitters in a simple polygon. The point 2-transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons.