Let $P$ be a set of points in the plane and let $m$ be an integer. The goal of Max Cover by Unit Disks problem is to place $m$ unit disks whose union covers the maximum number of points from~$P$. We are interested in the dynamic version of Max Cover by Unit Disks problem, where the points in $P$ appear and disappear over time, and the algorithm must maintain a set \cDalg of $m$ disks whose union covers many points. A dynamic algorithm for this problem is a $k$-stable $\alpha$-approximation algorithm when it makes at most $k$ changes to \cDalg upon each update to the set $P$ and the number of covered points at time $t$ is always at least $\alpha \cdot \opt(t)$, where $\opt(t)$ is the maximum number of points that can be covered by m disks at time $t$. We show that for any constant $\varepsilon>0$, there is a $k_{\varepsilon}$-stable $(1-\varepsilon)$-approximation algorithm for the dynamic Max Cover by Unit Disks problem, where $k_{\varepsilon}=O(1/\varepsilon^3)$. This improves the stability of $\Theta(1/\eps^4)$ that can be obtained by combining results of Chaplick, De, Ravsky, and Spoerhase (ESA 2018) and De~Berg, Sadhukhan, and Spieksma (APPROX 2023). Our result extends to other fat similarly-sized objects used in the covering, such as arbitrarily-oriented unit squares, or arbitrarily-oriented fat ellipses of fixed diameter. We complement the above result by showing that the restriction to fat objects is necessary to obtain a SAS. To this end, we study the Max Cover by Unit Segments problem, where the goal is to place $m$ unit-length segments whose union covers the maximum number of points from $P$. We show that there is a constant $\varepsilon^* > 0$ such that any $k$-stable $(1 + \varepsilon^*)$-approximation algorithm must have $k=\Omega(m)$, even when the point set never has more than four collinear points.

翻译：设 $P$ 为平面上的一组点集，$m$ 为整数。单位圆盘最大覆盖问题的目标是通过放置 $m$ 个单位圆盘，使其并集覆盖 $P$ 中尽可能多的点。我们关注该问题的动态版本：点集 $P$ 中的点会随时间动态出现或消失，算法需维护一个包含 $m$ 个圆盘的集合 \cDalg，使其并集持续覆盖大量点。若算法在点集 $P$ 每次更新时对 \cDalg 至多进行 $k$ 次修改，且任意时刻 $t$ 的覆盖点数始终不低于 $\alpha \cdot \opt(t)$（其中 $\opt(t)$ 表示时刻 $t$ 用 $m$ 个圆盘可覆盖的最大点数），则称该动态算法为 $k$-稳定 $\alpha$-逼近算法。我们证明对于任意常数 $\varepsilon>0$，动态单位圆盘最大覆盖问题存在 $k_{\varepsilon}$-稳定 $(1-\varepsilon)$-逼近算法，其中 $k_{\varepsilon}=O(1/\varepsilon^3)$。该结果将 Chaplick、De、Ravsky 与 Spoerhase（ESA 2018）以及 De~Berg、Sadhukhan 与 Spieksma（APPROX 2023）工作结合所能获得的稳定性界 $\Theta(1/\eps^4)$ 进行了改进。我们的结论可推广至其他具有相似尺寸的胖几何对象覆盖问题，例如任意朝向的单位正方形或固定直径的任意朝向胖椭圆。为说明对胖对象的限制是获得稳定逼近方案的必要条件，我们进一步研究单位线段最大覆盖问题：该问题要求放置 $m$ 条单位长度线段，使其并集覆盖 $P$ 中尽可能多的点。我们证明存在常数 $\varepsilon^* > 0$，使得任何 $k$-稳定 $(1 + \varepsilon^*)$-逼近算法必须满足 $k=\Omega(m)$，即使点集始终不存在超过四个共线点的情况亦然。