Fully Dynamic Geometric Vertex Cover and Matching

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}