Enclosing Points with Geometric Objects

Abstract

Let $X$ be a set of points in $\mathbb{R}^2$ and $\mathcal{O}$ be a set of geometric objects in $\mathbb{R}^2$, where $|X| + |\mathcal{O}| = n$. We study the problem of computing a minimum subset $\mathcal{O}^* \subseteq \mathcal{O}$ that encloses all points in $X$. Here a point $x \in X$ is enclosed by $\mathcal{O}^*$ if it lies in a bounded connected component of $\mathbb{R}^2 \backslash (\bigcup_{O \in \mathcal{O}^*} O)$. We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in $O(1)$-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an $O(α(n)\log n)$-approximation algorithm for segments, where $α(n)$ is the inverse Ackermann function, and an $O(\log n)$-approximation algorithm for disks.

Figures (4)

• Figure 1: The set of red disks enclose points $A$, $C$, $D$ and $E$, but does not enclose the point $B$.
• Figure 2: The winding numbers with respect to a (simple) polygon $P$. The point $q_1$ has winding number $c_{q_1}=1$, and is inside $P$; the point $q_2$ has winding number $c_{q_2}=0$, and is outside $P$.
• Figure 3: The outer boundary $\mathcal{B}$ of the union of segments is a set of disjoint simple polygons (shown in red).
• Figure 4: During the unwinding process, the non-simple cycle $C=u_1v_1w_2u_2v_2w_1$ with length 6 is decomposed into two simple cycles $C_1=vv_2w_1u_1$ and $C_2=vv_1w_2u_2$ each with length 4.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4