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 $n^{O(operatorname{poly}(loglog{n} / epsilon))}$. We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.
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