Guarding Offices with Maximum Dispersion

TL;DR

This paper studies a dispersive variant of the Art Gallery Problem in office-like polygons, proving NP-completeness results and providing polynomial-time algorithms for certain cases, with practical solver evaluations.

Contribution

It introduces complexity results for dispersion distances in office polygons and offers optimal algorithms for specific classes, extending previous open questions.

Findings

NP-completeness of achieving dispersion distance 4 in office polygons

Polynomial-time algorithm guaranteeing dispersion distance 3

Efficient SAT-based solvers for large instances with up to 1600 vertices

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.