Robustly Guarding Polygons

TL;DR

It is shown that imposing various degrees of robustness on the notion of visibility coverage leads to a more tractable (and realistic) problem for which to provide approximation algorithms with constant factor guarantees.

Abstract

We propose precise notions of what it means to guard a domain "robustly", under a variety of models. While approximation algorithms for minimizing the number of (precise) point guards in a polygon is a notoriously challenging area of investigation, we show that imposing various degrees of robustness on the notion of visibility coverage leads to a more tractable (and realistic) problem for which we can provide approximation algorithms with constant factor guarantees.

Figures (19)

• Figure 1: A point $g$ that $\alpha$-robustly guards another point $p$. The pink "ice cream cone" is contained in the polygon $P$.
• Figure 2: In a thin rectangle $P$, the point $g$$\alpha$-robustly guards $p_1$, but not $p_2$. Here, the number of robust guards required depends on the aspect ratio of the rectangle $P$.
• Figure 3: The cone $K$ with angle $\theta$. For $q$ on $\rho_{1/2}$, the disk $D(q,\frac{1}{2}\alpha\|p-q\|)$ is contained in $K$.
• Figure 4: The ice cream cone $C$ from $p$ to $g$ is shaded in pink. For any point $q \in C$, the ice cream cone from $q$ to $g$ is contained in $C$. Therefore, $\text{Vis}_\alpha(g)$ contains $C$.
• Figure 5: Computing $\text{Vis}_\alpha(g)$ as the intersection of heart shapes for every reflex vertex of $\text{Vis}(g)$. Left: the construction of a single heart shape (in violet). Right: $\text{Vis}_\alpha(g)$ is the area shaded in pink.

Theorems & Definitions (30)

• Definition 1: Robust Guarding
• Claim 3
• Lemma 4
• Lemma 5
• Claim 5
• Lemma 5
• Lemma 5
• Theorem 6
• Lemma 6
• Theorem 7