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  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) . In this paper, we present a 4-approximation algorithm for MISR which is based on a similar recursive partitioning scheme. However , we use a  more general class of polygons -- polygons that are horizontally or vertically convex -- which allows us to provide an arguably simpler analysis and already improve the approximation ratio . Using a new fractional charging argument and fork-fences to guide the partitions , we improve the approximation ratio even more to 4. We hope that our ideas will lead to further progress towards a PTAS for MISR.
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 , without intersecting certain special horizontal line segments called fences . In this paper, we present a 3-approximation algorithm for MISR which is based on a similar recursive partitioning scheme. First , we use a partition into a more general class of axis-parallel polygons with constant complexity each, which 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 3. In particular, our partitioning uses more general fences that can be sequences of up to O(1) line segments each. This and our other new ideas may be useful for future work towards a PTAS for MISR.
A
2021
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(CCRs)
, without intersecting certain special horizontal line segments called fences
R
4-approximation
3-approximation
R
However
First
A
partition into a
R
polygons -- polygons that are horizontally or vertically convex --
axis-parallel polygons with constant complexity each,
R
already improve
at the same time already improves
R
. Using a new fractional charging argument and fork-fences to guide the partitions
to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity
R
the approximation ratio even more to 4. We hope that our ideas will lead to further progress
our approximation ratio to 3. In particular, our partitioning uses more general fences that can be sequences of up to O(1) line segments each. This and our other new ideas may be useful for future work