A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem

TL;DR

This paper presents a deterministic bicriteria approximation algorithm for the art gallery problem, which efficiently finds a near-optimal set of points covering most of a polygon's area.

Contribution

It introduces a polynomial-time algorithm that guarantees a near-optimal solution with a provable approximation ratio for polygons with holes.

Findings

The algorithm achieves an approximation factor of O(OPT·log(h+2)·log(OPT·log(h+2)))

It guarantees coverage of at least (1−δ) of the polygon's area

The running time is polynomial in polygon complexity and 1/δ

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 $\delta \in(0,1)$, is guaranteed to find a set of points in $H$ of size $O\big(\OPT\cdot\log(h+2)\cdot\log (\OPT\cdot\log(h+2)))$ that sees at least a $(1-\delta)$-fraction of the area of the polygon. The running time of the algorithm is polynomial in $h$, $n$, $L$ and $\log(\frac{1}{\delta})$, where $\OPT$ is the size of an optimum solution.