The Contiguous Art Gallery Problem is Solvable in Polynomial Time

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 $\exists\mathbb{R}$-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 $\mathcal{O}\!\left( n^7 \log n \right)$. 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.