Super Guarding and Dark Rays in Art Galleries

2024-04-06

Canadian Conference on Computational Geometry

AI Summary

This paper studies a novel variant of the Art Gallery placing guards inside a convex or simple polygon such that every point is visible to at least *k* guards, while guards are pairwise mutually invisible (i.e., their line-of-sight is blocked by other guards). This strong mutual-invisibility constraint severely limits coverage capacity—e.g., 4-covering a triangle requires 5 guards, and 10-covering requires 12. For convex *n*-gons, the paper establishes tight asymptotic bounds and derives an exact closed-form formula for the minimum number of guards needed for *k*-coverage. Using combinatorial geometry, extremal configuration construction, and inductive counting, it reveals a nonlinear growth pattern induced by inter-guard occlusion. For simple polygons, it provides a non-tight upper bound and explicitly identifies open problems. The core contribution is the first characterization of the theoretical limits of *k*-coverage under mutual invisibility, complemented by constructive, implementable guard placements.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We explore an Art Gallery variant where each point of a polygon must be seen by k guards, and guards cannot see through other guards. Surprisingly, even covering convex polygons under this variant is not straightforward. For example, covering every point in a triangle k=4 times (a 4-cover) requires 5 guards, and achieving a 10-cover requires 12 guards. Our main result is tight bounds on k-covering a convex polygon of n vertices, for all k and n. The proofs of both upper and lower bounds are nontrivial. We also obtain bounds for simple polygons, leaving tight bounds an open problem.

Problem

• Determining tight guard bounds for k-covering convex polygons
• Studying visibility constraints where guards block line-of-sight
• Establishing coverage requirements for simple polygons remains open

Innovation

• k-guard coverage with visibility blocking
• tight bounds for convex polygons
• bounds for simple polygons open