Guarding Offices with Maximum Dispersion

2025-06-26

AI Summary

This paper studies the **maximum dispersion problem** for vertex guards in orthogonal polygons (modeling office floor plans): maximizing the L₁-geodesic distance—termed *dispersion distance*—between any two guards, subject to an *r*-visibility constraint, rather than minimizing guard count as in the classical art gallery problem. We first prove NP-completeness for dispersion distance ≥ 4. We then present worst-case optimal polynomial-time algorithms guaranteeing dispersion distances of 3 and 2, respectively. Our work resolves an open problem by Rieck and Scheffer on polyominoes. For hole-free, independent-office orthogonal polygons, we devise an exact dynamic programming algorithm. Integrating SAT/CP/MIP modeling with solver optimizations, we achieve optimal solutions within 15 seconds on random instances with up to 1600 vertices. Finally, we establish the NP-hardness threshold for rational-coordinate instances at dispersion distance 2 + ε.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We investigate the Dispersive Art Gallery Problem with vertex guards and rectangular visibility ($r$-visibility) for a class of orthogonal polygons that reflect the properties of real-world floor plans: these office-like polygons consist of rectangular rooms and corridors. In the dispersive variant of the Art Gallery Problem, the objective is not to minimize the number of guards but to maximize the minimum geodesic $L_1$-distance between any two guards, called the dispersion distance. Our main contributions are as follows. We prove that determining whether a vertex guard set can achieve a dispersion distance of $4$ in office-like polygons is NP-complete, where vertices of the polygon are restricted to integer coordinates. Additionally, we present a simple worst-case optimal algorithm that guarantees a dispersion distance of $3$ in polynomial time. Our complexity result extends to polyominoes, resolving an open question posed by Rieck and Scheffer (CGTA 2024). When vertex coordinates are allowed to be rational, we establish analogous results, proving that achieving a dispersion distance of $2+varepsilon$ is NP-hard for any $varepsilon > 0$, while the classic Art Gallery Problem remains solvable in polynomial time for this class of polygons. Furthermore, we give a straightforward polynomial-time algorithm that computes worst-case optimal solutions with a dispersion distance of $2$. On the other hand, for the more restricted class of hole-free independent office-like polygons, we propose a dynamic programming approach that computes optimal solutions. Moreover, we demonstrate that the problem is practically tractable for arbitrary orthogonal polygons. To this end, we compare solvers based on SAT, CP, and MIP formulations. Notably, SAT solvers efficiently compute optimal solutions for randomly generated instances with up to $1600$ vertices in under $15$s.

Problem

• Maximizing guard dispersion in office-like polygons
• Proving NP-completeness for dispersion distance of 4
• Providing polynomial-time algorithms for optimal dispersion

Innovation

• NP-complete proof for dispersion distance 4
• Polynomial algorithm for dispersion distance 3
• Dynamic programming for hole-free polygons