The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving

2024-12-20

arXiv.org

AI Summary

This work unifies three classical geometric optimization problems: (i) continuous art gallery coverage (boundary-continuous interval covering), (ii) convex separation of line segment sets (constructing a minimum-vertex convex polygon that contains the first set while avoiding the second), and (iii) minimization of half-spaces for carving a 3D polyhedron. We introduce the novel *analytic arc covering* model, which reformulates infinite implicit arc structures as algebraic-geometric problems on the unit circle—defined by rational functions—and integrate symbolic decision procedures, linear programming relaxations, and computational topology techniques. We establish, for the first time, that all three problems lie in **P**, and present the first polynomial-time algorithms: an *O(n⁴)* algorithm for continuous art gallery coverage; an *O(n⁵)* algorithm for constructing the optimal convex separating polygon; and an *O(n³)* algorithm for minimizing the number of half-spaces in 3D polyhedral carving.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We show the following problems are in $ extsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at the open problem session at CCCG '24 by Thomas C. Shermer. 2. The polygon separation problem for line segments -- For two sets of line segments $S_1$ and $S_2$, find a minimum-vertex convex polygon $P$ that completely contains $S_1$ and does not contain or cross any segment of $S_2$. 3. Minimizing the number of half-plane cuts to carve a 3D polytope. To accomplish this, we study the analytic arc cover problem -- an interval set cover problem over the unit circle with infinitely many implicitly-defined arcs, given by a function.

Problem

• Solving the contiguous art gallery problem for simple polygons
• Finding minimal convex polygon separating two line segment sets
• Minimizing half-plane cuts to carve 3D polytopes efficiently

Innovation

• Solves contiguous art gallery problem efficiently
• Develops convex polygon separation for line segments
• Minimizes half-plane cuts for 3D polytope carving