  We study the Maximum Independent Set of Rectangles (MISR) problem, where we
are given a set of axis-parallel rectangles in the plane and the goal is to
select a subset of non-overlapping rectangles of maximum cardinality. In a
recent breakthrough, Mitchell [2021] obtained the first constant-factor
approximation algorithm for MISR. His algorithm achieves an approximation ratio
of 10 and it is based on a dynamic program that intuitively recursively
partitions the input plane into special polygons called corner-clipped
rectangles (CCRs), without intersecting certain special horizontal line
segments called fences.
  In this paper, we present a $(2+\epsilon)$-approximation algorithm for MISR
which is also based on a recursive partitioning scheme. First, we use a
partition into a class of axis-parallel polygons with constant complexity each
that are more general than CCRs. This allows us to provide an arguably simpler
analysis and at the same time already improves the approximation ratio to 6.
Then, using a more elaborate charging scheme and a recursive partitioning into
general axis-parallel polygons with constant complexity, we improve our
approximation ratio to $2+\epsilon$. In particular, we construct a recursive
partitioning based on more general fences which can be sequences of up to
$O(1/\epsilon)$ line segments each. This partitioning routine and our other new
ideas may be useful for future work towards a PTAS for MISR.
