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.

翻译：我们研究的是最大独立矩形( MISR) 问题, 我们在这个问题上得到一组轴- 平行矩形( CCRS), 并且目标是选择一组非重叠的顶部矩形。 在最近的一个突破中, Mitchell [ 2021] 为 MISR 获得了第一个常数- 因素近似算法。 他的算法实现了10 的近似比率, 它基于一个动态程序, 该程序将输入的平面嵌入称为特殊多边形, 称为角- 圈矩形( CCRS), 而不将某些特殊的横向线形区段( 圈) 相互交叉, 目标是选择一个非重叠的矩形矩形。 我们在此文件中为 MISR 选择一个 $ ( 2 ⁇ eepslon) 的 组合矩形( $- proqol- proqual) 算法。 我们使用一个更精细的固定的平面阵列方案, 也可以用一个更精细的双向普通的双轴( ) 。