How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axis-parallel rectangles in the plane. Our goal is to compute a maximum weight subset of pairwise non-overlapping rectangles. Due to its various applications, as well as connections to many other problems in computer science, MWISR has received a lot of attention from the computational geometry and the approximation algorithms community. However, despite being extensively studied, MWISR
... ains not very well understood in terms of polynomial time approximation algorithms, as there is a large gap between the upper and lower bounds, i.e., O(log n/ log log n) v.s. NP-hardness. Another important, poorly understood question is whether one can color rectangles with at most O(ω(R)) colors where ω(R) is the size of a maximum clique in the intersection graph of a set of input rectangles R. Asplund and Grünbaum obtained an upper bound of O(ω(R) 2) about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR. In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of (1 − δ) for an arbitrarily small constant δ > 0.) Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with O(ω(R)) colors which implies a constant upper bound on the integrality gap of the canonical LP. For some applications of MWISR the possibility to shrink the rectangles has a natural, well-motivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.