In this work, we study two fundamental graph optimization problems, minimum vertex cover (MVC) and maximum-cardinality matching (MCM), for intersection graphs of geometric objects, e.g., disks, rectangles, hypercubes, etc., in $d$-dimensional Euclidean space. We consider the problems in fully dynamic settings, allowing insertions and deletions of objects. We develop a general framework for dynamic MVC in intersection graphs, achieving sublinear amortized update time for most natural families of geometric objects. In particular, we show that - \begin{itemize} \item For a dynamic collection of disks in $\mathbb{R}^2$ or hypercubes in $\mathbb{R}^d$ (for constant $d$), it is possible to maintain a $(1+\varepsilon)$-approximate vertex cover in $\polylog$ amortized update time. These results also hold in the bipartite case. \item For a dynamic collection of rectangles in $\mathbb{R}^2$, it is possible to maintain a $(\frac{3}{2}+\varepsilon)$-approximate vertex cover in $\polylog$ amortized update time. \end{itemize} Along the way, we obtain the first near-linear time static algorithms for MVC in the above two cases with the same approximation factors. Next, we turn our attention to the MCM problem. Although our MVC algorithms automatically allow us to approximate the size of the MCM in bipartite geometric intersection graphs, they do not produce a matching. We give another general framework to maintain an approximate maximum matching, and further extend the approach to handle non-bipartite intersection graphs. In particular, we show that - \begin{itemize} \item For a dynamic collection of (bichromatic or monochromatic) disks in $\mathbb{R}^2$ or hypercubes in $\mathbb{R}^d$ (for constant $d$), it is possible to maintain a $(1+\varepsilon)$-approximate matching in $\polylog$ amortized update time. \end{itemize}

翻译：在这项工作中，我们研究了两种基本的图优化问题——最小顶点覆盖（MVC）和最大基数匹配（MCM），针对$d$维欧几里得空间中几何对象的交集图（例如圆盘、矩形、超立方体等）。我们在完全动态设置中考虑这些问题，允许对象的插入和删除。我们为交集图上的动态MVC开发了一个通用框架，对于大多数自然类别的几何对象实现了次线性平摊更新时间。特别地，我们证明： - 对于$\mathbb{R}^2$中圆盘或$\mathbb{R}^d$中（常数$d$）超立方体的动态集合，可以在$\polylog$平摊更新时间内维护一个$(1+\varepsilon)$近似的顶点覆盖。这些结果也适用于二分图情况。 - 对于$\mathbb{R}^2$中矩形的动态集合，可以在$\polylog$平摊更新时间内维护一个$(\frac{3}{2}+\varepsilon)$近似的顶点覆盖。 在此过程中，我们首次获得了上述两种情况下MVC的近似线性时间静态算法，且具有相同的近似因子。接下来，我们将注意力转向MCM问题。尽管我们的MVC算法可以自动近似二分几何交集图中MCM的大小，但无法生成匹配。我们给出了另一个通用框架来维护近似最大匹配，并进一步扩展该方法以处理非二分交集图。特别地，我们证明： - 对于$\mathbb{R}^2$中（双色或单色）圆盘或$\mathbb{R}^d$中（常数$d$）超立方体的动态集合，可以在$\polylog$平摊更新时间内维护一个$(1+\varepsilon)$近似的匹配。