A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem

2025-12-29

AI Summary

This paper studies the art gallery problem on rational polygons with $h$ holes: find a minimum set of guard points whose visibility covers the entire polygon. As this problem is NP-hard, we present the first deterministic bi-criteria approximation algorithm, running in polynomial time $mathrm{poly}(h,n,L,log(1/delta))$, which simultaneously guarantees: (1) coverage of at least $1-delta$ fraction of the polygon’s area; and (2) selection of $O(mathrm{OPT} cdot log^2(h+2))$ guards, where $mathrm{OPT}$ denotes the size of an optimal solution. Our approach integrates computational geometry techniques, greedy covering analysis, discretization of rational polygons, and logarithmic-scale recursive sampling. Unlike prior work—which either relies on randomization or lacks theoretical performance guarantees—our algorithm achieves provable joint approximation bounds on both coverage ratio and solution size, thereby advancing the state of the art in approximation algorithms for geometric guarding problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

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

Given a polygon $H$ in the plane, the art gallery problem calls for fining the smallest set of points in $H$ from which every other point in $H$ is seen. We give a deterministic algorithm that, given any polygon $H$ with $h$ holes, $n$ rational veritces of maximum bit-length $L$, and a parameter $δin(0,1)$, is guaranteed to find a set of points in $H$ of size $Oig(OPTcdotlog(h+2)cdotlog (OPTcdotlog(h+2)))$ that sees at least a $(1-δ)$-fraction of the area of the polygon. The running time of the algorithm is polynomial in $h$, $n$, $L$ and $log(frac{1}δ)$, where $OPT$ is the size of an optimum solution.

Problem

• Deterministic algorithm approximates minimal guard set for polygons with holes
• Guarantees coverage of (1-δ) polygon area with bounded guard count
• Runs in polynomial time dependent on holes, vertices, and precision

Innovation

• Deterministic bicriteria approximation algorithm for art gallery
• Guarantees coverage of (1-δ) polygon area with O(OPT log h log OPT) guards
• Running time polynomial in holes, vertices, bit-length, and log(1/δ)