Smoothed Analysis of the Art Gallery Problem

TL;DR

This paper applies smoothed analysis to the Art Gallery Problem, showing that in typical cases, optimal guard positions have low complexity and algebraic methods are unnecessary, providing a theoretical explanation for the rarity of irrational guard coordinates.

Contribution

It introduces the first smoothed analysis of the Art Gallery Problem, demonstrating that optimal guard positions typically have low bit-complexity under small perturbations.

Findings

Expected bits to describe guard positions are logarithmic in input size and perturbation magnitude.

Rational guard positions with small bit-complexity are typical in smoothed instances.

Algebraic methods are unnecessary for solving typical instances of the problem.

Abstract

In the Art Gallery Problem we are given a polygon $P\subset [0,L]^2$ on $n$ vertices and a number $k$. We want to find a guard set $G$ of size $k$, such that each point in $P$ is seen by a guard in $G$. Formally, a guard $g$ sees a point $p \in P$ if the line segment $pg$ is fully contained inside the polygon $P$. The history and practical findings indicate that irrational coordinates are a "very rare" phenomenon. We give a theoretical explanation. Next to worst case analysis, Smoothed Analysis gained popularity to explain the practical performance of algorithms, even if they perform badly in the worst case. The idea is to study the expected performance on small perturbations of the worst input. The performance is measured in terms of the magnitude $\delta$ of the perturbation and the input size. We consider four different models of perturbation. We show that the expected number of bits to describe optimal guard positions per guard is logarithmic in the input and the magnitude of the perturbation. This shows from a theoretical perspective that rational guards with small bit-complexity are typical. Note that describing the guard position is the bottleneck to show NP-membership. The significance of our results is that algebraic methods are not needed to solve the Art Gallery Problem in typical instances. This is the first time an $\exists\mathbb{R}$-complete problem was analyzed by Smoothed Analysis.