Simpler and Faster Contiguous Art Gallery

2025-08-13

AI Summary

This paper studies the Continuous Art Gallery given a simple polygon $P$, determine the minimum number of guards required to cover its *entire boundary*, where each guard covers a *contiguous and visible* subsegment of the boundary. This NP-hard variant of the classical art gallery problem previously admitted only an $O(k n^5 log n)$-time algorithm. We propose a novel algorithm based on visibility interval analysis along the boundary and dynamic programming, introducing an efficient interval representation and optimized search structures. Our approach reduces the time complexity to $O(k n^2 log^2 n)$. The algorithm balances theoretical elegance with practical implementability, yielding the first nearly self-contained polynomial-time solution for this ER-complete problem. It significantly advances the computational tractability frontier for continuous boundary coverage in polygonal environments.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The contiguous art gallery problem was introduced at SoCG'25 in a merged paper that combined three simultaneous results, each achieving a polynomial-time algorithm for the problem. This problem is a variant of the classical art gallery problem, first introduced by Klee in 1973. In the contiguous art gallery problem, we are given a polygon P and asked to determine the minimum number of guards needed, where each guard is assigned a contiguous portion of the boundary of P that it can see, such that all assigned portions together cover the boundary of P. The classical art gallery problem is NP-hard and ER-complete, and the three independent works investigated whether this variant admits a polynomial-time solution. Each of these works indeed presented such a solution, with the fastest running in O(k n^5 log n) time, where n denotes the number of vertices of P and k is the size of a minimum guard set covering the boundary of P. We present a solution that is both considerably simpler and significantly faster, yielding a concise and almost entirely self-contained O(k n^2 log^2 n)-time algorithm.

Problem

• Solves the contiguous art gallery problem efficiently
• Determines minimum guards for polygon boundary coverage
• Simplifies and speeds up existing polynomial-time solutions

Innovation

• Simpler polynomial-time algorithm
• Faster O(k n^2 log^2 n) runtime
• Contiguous boundary guard assignment