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 + ε)-approximation algorithm for the MWISR problem with quasipolynomial running time 2
^{poly(log n/ε)}
. 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. [4] this allows us to upper bound our approximation ratio by 1 + ε. 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 + ε)-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. In particular, note that our results imply that the MWISR problem is not APX-hard, unless NP ⊆ DTIME(2
^{polylog (n)}
).