  In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are
given a weighted set of $n$ axis-parallel rectangles in the plane. The task is
to find a subset of pairwise non-overlapping rectangles with the maximum
possible total weight. This problem is NP-hard and the best-known
polynomial-time approximation algorithm, due to by Chalermsook and Walczak
(SODA 2021), achieves approximation factor $O(\log\log n )$. While in the
unweighted setting, constant factor approximation algorithms are known, due to
Mitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open to
extend these techniques to the weighted setting.
  In this paper, we consider MWISR through the lens of parameterized
approximation. Grandoni et al. (ESA 2019) gave a $(1-\epsilon)$-approximation
algorithm with running time $k^{O(k/\epsilon^8)} n^{O(1/\epsilon^8)}$ time,
where $k$ is the number of rectangles in an optimum solution. Unfortunately,
their algorithm works only in the unweighted setting and they left it as an
open problem to give a parameterized approximation scheme in the weighted
setting.
  Our contribution is a partial answer to the open question of Grandoni et al.
(ESA 2019). We give a parameterized approximation algorithm for MWISR that
given a parameter $k$, finds a set of non-overlapping rectangles of weight at
least $(1-\epsilon) \text{opt}_k$ in $2^{O(k \log(k/\epsilon))}
n^{O(1/\epsilon)}$ time, where $\text{opt}_k$ is the maximum weight of a
solution of cardinality at most $k$. Note that thus, our algorithm may return a
solution consisting of more than $k$ rectangles. To complement this apparent
weakness, we also propose a parameterized approximation scheme with running
time $2^{O(k^2 \log(k/\epsilon))} n^{O(1)}$ that finds a solution with
cardinality at most $k$ and total weight at least $(1-\epsilon)\text{opt}_k$
for the special case of axis-parallel segments.
