Guarding and Searching Polyhedra

We tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra.

💡 Research Summary

This doctoral thesis, “Guarding and Searching Polyhedra,” presents a comprehensive investigation into the computational geometry of surveillance and pursuit-evasion within three-dimensional polyhedral environments. It focuses on two classical problems: the static Art Gallery Problem and the dynamic Searchlight Scheduling Problem, extending them from their well-studied 2D polygonal domains to 3D.

The research is structured in three main parts. Part I establishes foundational definitions for polyhedra (distinguishing between “closed” and “open” models), introduces various guard types (point, edge, face), and reviews relevant 2D literature.

Part II is dedicated to the Art Gallery Problem using edge guards, argued to be the most natural 3D analogue of vertex guards in polygons. The core results here involve establishing tight bounds on the number of guards required for specific classes of polyhedra. A key contribution is a generalization of O’Rourke’s classic theorem to 3D: for orthogonal polyhedra whose reflex edges lie in only two directions (2-reflex orthogonal polyhedra), ⌊r/2⌋ + 1 reflex edge guards are sufficient and sometimes necessary, where r is the number of reflex edges. This guard set can be computed in O(n log n) time. For general orthogonal polyhedra guarded by mutually parallel edge guards, the thesis improves the previously best-known upper bound from ⌊e/6⌋ to ⌊11e/72⌋, where e is the total number of edges, and provides a linear-time algorithm. Tight inequalities relating e and r are also derived, yielding an upper bound of ⌊7r/12⌋ + 1. Furthermore, the thesis investigates polyhedra with faces oriented in only four directions, establishing a lower bound of ⌊e/6⌋ - 1 and an upper bound of ⌊(e+r)/6⌋ edge guards. All these bounds hold for polyhedra of any genus.

Part III formulates and analyzes the 3D Searchlight Scheduling Problem. The author proposes a model where guards are line segments that rotate a half-plane of light (a “searcherplane”) around their axis. After defining the model and discussing special guard types like “filling guards” (which enable a 3D generalization of the one-way sweep strategy), the thesis tackles several complexity-theoretic questions. It is shown that placing at most r² guards suffices to make any polyhedron searchable (reducing to r guards for orthogonal polyhedra). However, deciding if a given set of guards can search the entire polyhedron is proven to be strongly NP-hard. Even under the promise that an orthogonal polyhedron is searchable, approximating the minimum search time within a constant factor is also strongly NP-hard. A significant breakthrough is achieved for the “Partial Searchlight Scheduling Problem,” where only a specified subregion needs to be cleared. This problem is shown to be strongly PSPACE-hard for 3D polyhedra. By refining the construction, the author proves that the 2D version of this partial search problem for orthogonal polygons is strongly PSPACE-complete. This latter result is particularly noteworthy as it provides the first strong complexity lower bound (PSPACE-completeness) for any variant of the 2D Searchlight Scheduling Problem.

In summary, the thesis makes substantial contributions by providing new combinatorial bounds for guarding 3D polyhedra, formalizing a 3D search model, and establishing a series of hardness results that reveal the intrinsic computational difficulty of these problems, even in restricted settings like orthogonal polyhedra and polygons.