  We present an $(1+\varepsilon)$-approximation algorithm with quasi-polynomial
running time for computing the maximum weight independent set of polygons out
of a given set of polygons in the plane (specifically, the running time is
$n^{O( \mathrm{poly}( \log n, 1/\varepsilon))}$). Contrasting this, the best
known polynomial time algorithm for the problem has an approximation ratio
of~$n^{\varepsilon}$. Surprisingly, we can extend the algorithm to the problem
of computing the maximum weight subset of the given set of polygons whose
intersection graph fulfills some sparsity condition. For example, we show that
one can approximate the maximum weight subset of polygons, such that the
intersection graph of the subset is planar or does not contain a cycle of
length $4$ (i.e., $K_{2,2}$). Our algorithm relies on a recursive partitioning
scheme, whose backbone is the existence of balanced cuts with small complexity
that intersect polygons from the optimal solution of a small total weight.
  For the case of large axis-parallel rectangles, we provide a polynomial time
$(1+\varepsilon)$-approximation for the maximum weight independent set.
Specifically, we consider the problem where each rectangle has one edge whose
length is at least a constant fraction of the length of the corresponding edge
of the bounding box of all the input elements. This is now the most general
case for which a PTAS is known, and it requires a new and involved partitioning
scheme, which should be of independent interest.
