  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.
