Algorithms for Art Gallery Illumination

TL;DR

This paper extends the classical Art Gallery Problem to include fading light intensity, presenting complexity results and practical algorithms for fixed guard positions that effectively minimize energy consumption in illumination.

Contribution

It introduces the AGPF, analyzes its complexity, and proposes two practical algorithms that work well in real-world scenarios for fixed guard positions.

Findings

The AGPF is NP-hard with negative solvability results.

A fully polynomial-time approximation scheme is developed for fixed light positions.

The non-linear programming approach yields superior practical results.

Abstract

The Art Gallery Problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the Art Gallery Problem with Fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation.