Searching Polyhedra by Rotating Half-Planes

TL;DR

This paper extends the 2D searchlight scheduling problem to 3D polyhedra, establishing guard placement bounds, complexity results, and demonstrating the problem's computational hardness.

Contribution

It introduces a 3D searchlight scheduling model, extends planar strategies, and proves NP-hardness and PSPACE-hardness for key decision problems.

Findings

Polyhedra with r reflex edges can be searched with at most r^2 guards.

Orthogonal polyhedra can be searched with r guards.

Deciding searchability and schedule existence is NP-hard and PSPACE-hard.

Abstract

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.