4 research outputs found
Approximation Schemes for Maximum Weight Independent Set of Rectangles
In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are
given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to
select a maximum weight subset of pairwise non-overlapping rectangles. Due to
many applications, e.g. in data mining, map labeling and admission control, the
problem has received a lot of attention by various research communities. We
present the first (1+epsilon)-approximation algorithm for the MWISR problem
with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the
best known polynomial time approximation algorithms for the problem achieve
superconstant approximation ratios of O(log log n) (unweighted case) and O(log
n / log log n) (weighted case).
Key to our results is a new geometric dynamic program which recursively
subdivides the plane into polygons of bounded complexity. We provide the
technical tools that are needed to analyze its performance. In particular, we
present a method of partitioning the plane into small and simple areas such
that the rectangles of an optimal solution are intersected in a very controlled
manner. Together with a novel application of the weighted planar graph
separator theorem due to Arora et al. this allows us to upper bound our
approximation ratio by (1+epsilon).
Our dynamic program is very general and we believe that it will be useful for
other settings. In particular, we show that, when parametrized properly, it
provides a polynomial time (1+epsilon)-approximation for the special case of
the MWISR problem when each rectangle is relatively large in at least one
dimension. Key to this analysis is a method to tile the plane in order to
approximately describe the topology of these rectangles in an optimal solution.
This technique might be a useful insight to design better polynomial time
approximation algorithms or even a PTAS for the MWISR problem
A 3-Approximation Algorithm for Maximum Independent Set of Rectangles
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.Comment: 41 page