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 $O(\alpha(n)\log n)$-approximation algorithm for segments, where $\alpha(n)$ is the inverse Ackermann function, and an $O(\log n)$-approximation algorithm for disks.

翻译：设 $X$ 为 $\mathbb{R}^2$ 中的点集，$\mathcal{O}$ 为 $\mathbb{R}^2$ 中的几何对象集，满足 $|X| + |\mathcal{O}| = n$。我们研究计算最小子集 $\mathcal{O}^* \subseteq \mathcal{O}$ 以包围 $X$ 中所有点的问题。此处，点 $x \in X$ 被 $\mathcal{O}^*$ 包围，当且仅当它位于 $\mathbb{R}^2 \backslash (\bigcup_{O \in \mathcal{O}^*} O)$ 的一个有界连通分量内。我们提出两种算法框架来设计该问题的多项式时间近似算法。第一种框架基于稀疏化和最小割，可为单位圆盘、单位正方形等情形给出 $O(1)$ 近似算法。第二种框架基于线性规划舍入，可为线段情形给出 $O(\alpha(n)\log n)$ 近似算法（其中 $\alpha(n)$ 为逆阿克曼函数），并为圆盘情形给出 $O(\log n)$ 近似算法。