  We study the Maximum Independent Set of Rectangles (MISR) problem: given a
set of $n$ axis-parallel rectangles, find a largest-cardinality subset of the
rectangles, such that no two of them overlap. MISR is a basic geometric
optimization problem with many applications, that has been studied extensively.
Until recently, the best approximation algorithm for it achieved an $O(\log
\log n)$-approximation factor. In a recent breakthrough, Adamaszek and Wiese
provided a quasi-polynomial time approximation scheme: a
$(1-\epsilon)$-approximation algorithm with running time
$n^{O(\operatorname{poly}(\log n)/\epsilon)}$. Despite this result, obtaining a
PTAS or even a polynomial-time constant-factor approximation remains a
challenging open problem. In this paper we make progress towards this goal by
providing an algorithm for MISR that achieves a $(1 - \epsilon)$-approximation
in time $n^{O(\operatorname{poly}(\log\log{n} / \epsilon))}$. We introduce
several new technical ideas, that we hope will lead to further progress on this
and related problems.
