Approximation Schemes for Maximum Weight Independent Set of Rectangles

Abstract

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 quasi-polynomial 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. [3] 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) ). *

Independent Set of Convex Polygons: From $$n^{\epsilon }$$ n ϵ to $$1+\epsilon $$ 1 + ϵ

“…Going beyond polynomial time results, for independent set of arbitrary polygons there is a QPTAS [3,17], i.e., a (1 + )-approximation in time n (log n) O (1) , building on an earlier QPTAS for axis-parallel rectangles [2]. This implies that all the above problems are not APX-hard, unless NP ⊆ DTIME(n poly(log n) ).…”

“…Then on p 1 the path Q 1 hits a polygon P ∈ P i . Thus, the boundary of P must intersect (2) at a point p. Then the y-coordinate of this point p must be lower than the y-coordinate of p 2 (since Q 1 goes monotonely upwards). Then we set P 1 := P and A 1 consists of the quadrilateral described by p 1 , p 1 , p, p 2 , and…”

“…There is no point on the boundary of Aṽ in which a grid line of G i intersects P 1 . Thus, for the points p 1 and p 2 on which the paths Q 1 and Q 2 start, there must be two consecutive grid lines (1) , (2) of G i such that (1) intersects P 1 on p 1 and (2) intersects P 1 on p 2 , see Fig. 5.…”

“…5. Assume that (1) is on the left of (2) . Since Aṽ is not contained in a grid column of G i one of the paths Q 1 , Q 2 is not completely vertical.…”

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

“…Baker's approach was extended by Eppstein [Epp00] to graphs with bounded local treewidth, and by Grohe [Gro03] to graphs excluding minors. Separators have also played a key role in geometric optimization algorithms, including: (i) PTAS for independent set and (continuous) piercing set for fat objects [Cha03,MR10], (ii) QPTAS for maximum weighted independent sets of polygons [AW13,AW14,Har14], and (iii) QPTAS for Set Cover by pseudodisks [MRR14a], among others. Lastly, Cabello and Gajser [CG14a] develop PTAS's for some of the problems we study in the specific setting of minor-free graphs.…”