The Contiguous Art Gallery Problem is Solvable in Polynomial Time

TL;DR

This paper proves that the contiguous art gallery problem, involving partitioning a polygon boundary into visible chains, can be solved in polynomial time using a practical greedy algorithm, contrasting with many NP-hard variants.

Contribution

The paper introduces the first polynomial-time algorithm for the contiguous art gallery problem, providing a simple, practical solution and implementation, unlike other variants that are NP-hard or $orall ext{-} ext{R}$-complete.

Findings

Polynomial-time algorithm for the contiguous art gallery problem.

Algorithm runs in $ ext{O}(n^6 ext{log} n)$ time.

Provides bounds on bit complexity and algorithms for restricted cases.

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 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 classical art gallery problem and the edge-covering problem, both of which being proven to be $\exists\mathbb{R}$-complete recently by Abrahamsen, Adamaszek, and Miltzow [J. ACM 2022] and Stade [SoCG 2025], respectively. 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 $\mathcal{O}\!\left( n^6 \log n \right)$ arithmetic operations. We further bound the bit complexity of the computed values, showing that problem is in P. 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.