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 - - 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. - 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. 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 - - 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.

翻译：本研究针对$d$维欧氏空间中几何对象（如圆盘、矩形、超立方体等）的交图，探讨了两个基础图优化问题：最小顶点覆盖（MVC）与最大基数匹配（MCM）。我们考虑全动态环境下的问题，允许对象进行插入和删除操作。针对交图中的动态MVC问题，我们构建了一个通用框架，对大多数自然几何对象族实现了次线性均摊更新复杂度。具体而言，我们证明了： - 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，可以在多对数均摊更新时间内维护$(1+\varepsilon)$近似顶点覆盖；该结论同样适用于二分图情形。 - 对于$\mathbb{R}^2$中矩形的动态集合，可以在多对数均摊更新时间内维护$(\frac{3}{2}+\varepsilon)$近似顶点覆盖。 在这一过程中，我们首次针对上述两类图，获得了具有相同近似比的多项式时间静态MVC算法。随后，我们关注MCM问题。尽管上述MVC算法能自动估算二分几何交图中MCM的规模，但无法直接生成匹配结果。为此，我们提出了另一个维护近似最大匹配的通用框架，并将其扩展至非二分交图。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，可以在多对数均摊更新时间内维护$(1+\varepsilon)$近似匹配。