We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set $S$ of points and a set $\mathcal{R}$ of geometric objects, and the goal is to find a subset $\mathcal{R}^*\subseteq\mathcal{R}$ to cover all points in $S$ such that the \textit{membership} of $S$ with respect to $\mathcal{R}^*$, denoted by $\mathsf{memb}(S,\mathcal{R}^*)$, is minimized, where $\mathsf{memb}(S,\mathcal{R}^*)=\max_{p\in S}|\{R\in\mathcal{R}^*: p\in R\}|$. We achieve the following two main results. * We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time $n^{O(\mathsf{opt})}$ where $\mathsf{opt}$ is the optimum of the problem (i.e., the minimum membership). * We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of $o(\log n)$ in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find $\mathcal{R}^*\subseteq\mathcal{R}$ to cover $S$ such that the ply of $\mathcal{R}^*$ is minimized, where the ply is defined as the maximum number of objects in $\mathcal{R}^*$ which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in $O(n^{12})$ time. We give a significantly simpler constant-approximation algorithm with near-linear running time.

翻译：我们重新探讨了一种几何集覆盖的自然变体，称为最小成员几何集覆盖（MMGSC）。在该问题中，输入由一个点集$S$和一个几何对象集$\mathcal{R}$组成，目标是从$\mathcal{R}$中选取子集$\mathcal{R}^*\subseteq\mathcal{R}$覆盖$S$中的所有点，使得$S$关于$\mathcal{R}^*$的\textit{成员度}（记为$\mathsf{memb}(S,\mathcal{R}^*)$）最小化，其中$\mathsf{memb}(S,\mathcal{R}^*)=\max_{p\in S}|\{R\in\mathcal{R}^*: p\in R\}|$。我们取得了以下两个主要结果： * 针对单位正方形情形，我们首次给出了MMGSC的多项式时间常数近似算法。这回答了自Erlebach和Leeuwen [SODA'08] 工作以来遗留的公开问题，他们曾给出运行时间为$n^{O(\mathsf{opt})}$的常数近似算法，其中$\mathsf{opt}$是问题的最优值（即最小成员度）。 * 针对半平面情形，我们首次给出了MMGSC的多项式时间近似方案（PTAS）。在此工作之前，甚至无法确定该问题是否能在多项式时间内实现$o(\log n)$的近似因子，而众所周知半平面情形下的最小基数集覆盖问题可在多项式时间内精确求解。 我们还考虑了一个与MMGSC密切相关的问题，称为最小层数几何集覆盖（MPGSC），其目标是在$\mathcal{R}$中选取子集$\mathcal{R}^*\subseteq\mathcal{R}$覆盖$S$，使得$\mathcal{R}^*$的层数最小化，其中层数定义为$\mathcal{R}^*$中具有非空公共交集的对象的最大数量。最近，Durocher等人首次给出了运行时间为$O(n^{12})$的单位正方形MPGSC常数近似算法。我们提供了一个显著更简单的、具有近线性运行时间的常数近似算法。