完全动态几何顶点覆盖与匹配
Fully Dynamic Geometric Vertex Cover and Matching
AI总结：
针对动态几何交集图，提出了在多对数均摊更新时间内维护最小顶点覆盖近似和最大基数匹配近似的通用框架，并取得了首个近线性时间静态算法。
摘要：
在本研究中，我们研究了两个基本的图优化问题：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（例如圆盘、矩形、超立方体等）的交集图。我们在完全动态环境下考虑这些问题，允许对象的插入和删除。我们为交集图中的动态 MVC 开发了一个通用框架，在大多数自然几何对象族中实现了次线性的均摊更新时间。具体而言，我们证明了：- 对于 $\mathbb{R}^2$ 中的动态圆盘集合或 $\mathbb{R}^d$（常数 $d$）中的超立方体集合，可以在多对数均摊更新时间内维护 $(1+\varepsilon)$-近似顶点覆盖。这些结果在二分图情况下同样成立。- 对于 $\mathbb{R}^2$ 中的动态矩形集合，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。在此过程中，我们针对上述两种情况获得了首个具有相同近似因子的近线性时间静态 MVC 算法。接下来，我们将注意力转向 MCM 问题。尽管我们的 MVC 算法自动允许我们在二分几何交集图中近似 MCM 的大小，但它们并不产生匹配。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。具体而言，我们证明了：- 对于 $\mathbb{R}^2$ 中的动态（双色或单色）圆盘集合或 $\mathbb{R}^d$（常数 $d$）中的超立方体集合，可以在多对数均摊更新时间内维护 $(1+\varepsilon)$-近似匹配。
英文摘要：
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.
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