Bandyapadhyay et al. introduced the generalized minimum-membership geometric set cover (GMMGSC) problem [SoCG, 2023], which is defined as follows. We are given two sets $P$ and $P'$ of points in $\mathbb{R}^{2}$, $n=\max(|P|, |P'|)$, and a set $\mathcal{S}$ of $m$ axis-parallel unit squares. The goal is to find a subset $\mathcal{S}^{*}\subseteq \mathcal{S}$ that covers all the points in $P$ while minimizing $\mathsf{memb}(P', \mathcal{S}^{*})$, where $\mathsf{memb}(P', \mathcal{S}^{*})=\max_{p\in P'}|\{s\in \mathcal{S}^{*}: p\in s\}|$. We study GMMGSC problem and give a $16$-approximation algorithm that runs in $O(m^2\log m + m^2n)$ time. Our result is a significant improvement to the $144$-approximation given by Bandyapadhyay et al. that runs in $\tilde{O}(nm)$ time. GMMGSC problem is a generalization of another well-studied problem called Minimum Ply Geometric Set Cover (MPGSC), in which the goal is to minimize the ply of $\mathcal{S}^{*}$, where the ply is the maximum cardinality of a subset of the unit squares that have a non-empty intersection. The best-known result for the MPGSC problem is an $8$-approximation algorithm by Durocher et al. that runs in $O(n + m^{8}k^{4}\log k + m^{8}\log m\log k)$ time, where $k$ is the optimal ply value [WALCOM, 2023].

翻译：Bandyapadhyay等人在[SoCG, 2023]中提出了广义最小隶属度几何集合覆盖（GMMGSC）问题，其定义如下：给定$\mathbb{R}^{2}$中的两组点集$P$和$P'$（$n=\max(|P|, |P'|)$），以及一个由$m$个轴对齐单位正方形构成的集合$\mathcal{S}$。目标是找到子集$\mathcal{S}^{*}\subseteq \mathcal{S}$，使其覆盖$P$中所有点，同时最小化$\mathsf{memb}(P', \mathcal{S}^{*})$，其中$\mathsf{memb}(P', \mathcal{S}^{*})=\max_{p\in P'}|\{s\in \mathcal{S}^{*}: p\in s\}|$。我们研究GMMGSC问题并给出一个时间复杂度为$O(m^2\log m + m^2n)$的16-近似算法。该结果显著改进了Bandyapadhyay等人提出的144-近似算法（运行时间为$\tilde{O}(nm)$）。GMMGSC问题是另一个被广泛研究的问题——最小堆积几何集合覆盖（MPGSC）的推广形式，MPGSC问题的目标是最小化$\mathcal{S}^{*}$的堆积数（即具有非空交集的最大单位正方形子集基数）。目前MPGSC问题的最佳已知结果是Durocher等人提出的8-近似算法，其时间复杂度为$O(n + m^{8}k^{4}\log k + m^{8}\log m\log k)$，其中$k$为最优堆积值[WALCOM, 2023]。