The Contiguous Art Gallery Problem is Solvable in Polynomial Time

2024-12-18

arXiv.org

AI Summary

This paper studies the Contiguous Art Gallery given a simple polygon $P$, partition its boundary into the minimum number of contiguous chains such that each chain is fully visible from a single guard—placed anywhere on or inside $P$. We establish, for the first time, that this problem is solvable to optimality in polynomial time under the real RAM model, breaking the common theoretical barrier where most art gallery variants are NP-hard or $existsmathbb{R}$-complete. We propose a deterministic greedy algorithm with time complexity $O(n^7 log n)$, which also handles the vertex-constrained variant (where guards and chain endpoints must be vertices). The algorithm computes an optimal partition via a finite number of boundary traversals. An open-source C++ implementation is provided. This work identifies the contiguous variant as a rare polynomial-time-solvable instance within computational geometry’s art gallery problem family.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

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

In this paper, we study the Contiguous Art Gallery Problem, introduced by Thomas C. Shermer at the 2024 Canadian Conference on Computational Geometry, a variant of the classical art gallery problem from 1973 by Victor Klee. In the contiguous variant, the input is a simple polygon $P$, and the goal is to partition the boundary into a minimum number of polygonal chains such that each chain is visible to a guard. We present a polynomial-time real RAM algorithm, which solves the contiguous art gallery problem. Our algorithm is simple and practical, and we make a C++ implementation available. In contrast, many variations of the art gallery problem are at least NP-hard, making the contiguous variant stand out. These include the edge-covering problem, proven NP-hard by Laurentini [The Visual Computer 1999], and the classical art gallery problem, recently shown $existsmathbb{R}$-complete by Abrahamsen, Adamaszek, and Miltzow [J. ACM 2022]. Our algorithm is a greedy algorithm that repeatedly traverses the polygon's boundary. To find an optimal solution, we show that it is sufficient to traverse the polygon polynomially many times, resulting in a runtime of $mathcal{O}!left( n^7 log n ight)$. Additionally, we provide algorithms for the restricted settings, where either the endpoints of the polygonal chains or the guards must coincide with the vertices of the polygon.

Problem

• Solves the Contiguous Art Gallery Problem efficiently
• Minimizes polygonal chains for guard visibility
• Provides polynomial-time algorithm for boundary partitioning

Innovation

• Polynomial-time real RAM algorithm
• Greedy boundary traversal approach
• C++ implementation for practicality