In the Maximum Independent Set of Objects problem, we are given an $n$-vertex planar graph $G$ and a family $\mathcal{D}$ of $N$ objects, where each object is a connected subgraph of $G$. The task is to find a subfamily $\mathcal{F} \subseteq \mathcal{D}$ of maximum cardinality that consists of pairwise disjoint objects. This problem is $\mathsf{NP}$-hard and is equivalent to the problem of finding the maximum number of pairwise disjoint polygons in a given family of polygons in the plane. As shown by Adamaszek et al. (J. ACM '19), the problem admits a \emph{quasi-polynomial time approximation scheme} (QPTAS): a $(1-\varepsilon)$-approximation algorithm whose running time is bounded by $2^{\mathrm{poly}(\log(N),1/\epsilon)} \cdot n^{\mathcal{O}(1)}$. Nevertheless, to the best of our knowledge, in the polynomial-time regime only the trivial $\mathcal{O}(N)$-approximation is known for the problem in full generality. In the restricted setting where the objects are pseudolines in the plane, Fox and Pach (SODA '11) gave an $N^{\varepsilon}$-approximation algorithm with running time $N^{2^{\tilde{\mathcal{O}}(1/\varepsilon)}}$, for any $\varepsilon>0$. In this work, we present an $\text{OPT}^{\varepsilon}$-approximation algorithm for the problem that runs in time $N^{\tilde{\mathcal{O}}(1/\varepsilon^2)} n^{\mathcal{O}(1)}$, for any $\varepsilon>0$, thus improving upon the result of Fox and Pach both in terms of generality and in terms of the running time. Our approach combines the methodology of Voronoi separators, introduced by Marx and Pilipczuk (TALG '22), with a new analysis of the approximation factor.

翻译：在最大对象独立集问题中，给定一个 $n$ 顶点平面图 $G$ 和一个由 $N$ 个对象组成的族 $\mathcal{D}$，其中每个对象是 $G$ 的连通子图。任务是找到一个由两两不相交对象组成的最大基数子族 $\mathcal{F} \subseteq \mathcal{D}$。该问题是 $\mathsf{NP}$-困难的，等价于在平面给定多边形族中寻找最大数量的两两不相交多边形。如 Adamaszek 等人 (J. ACM '19) 所示，该问题存在一个拟多项式时间近似方案 (QPTAS)：一个 $(1-\varepsilon)$-近似算法，其运行时间受 $2^{\mathrm{poly}(\log(N),1/\epsilon)} \cdot n^{\mathcal{O}(1)}$ 界限制。然而，据我们所知，在多项式时间范围内，该问题在完全一般性下仅已知平凡 $\mathcal{O}(N)$-近似。在对象为平面伪直线的受限设定下，Fox 和 Pach (SODA '11) 给出了一个 $N^{\varepsilon}$-近似算法，运行时间为 $N^{2^{\tilde{\mathcal{O}}(1/\varepsilon)}}$，适用于任意 $\varepsilon>0$。在本工作中，我们针对该问题提出一个 $\text{OPT}^{\varepsilon}$-近似算法，运行时间为 $N^{\tilde{\mathcal{O}}(1/\varepsilon^2)} n^{\mathcal{O}(1)}$，适用于任意 $\varepsilon>0$，从而在一般性和运行时间两方面均改进了 Fox 和 Pach 的结果。我们的方法结合了 Marx 和 Pilipczuk (TALG '22) 引入的 Voronoi 分隔符方法论，以及对近似因子的新分析。