A 3-Approximation Algorithm for Maximum Independent Set of Rectangles Waldo Gálvez * Arindam Khan † Mathieu Mari ‡ Tobias Mömke § Madhusudhan Reddy Pittu ¶ Andreas Wiese || Abstract 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 [46] 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), without intersecting certain special horizontal line segments called fences. In this paper, we present a 3-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This 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, we construct a recursive partitioning based on more general fences which can be sequences of up to O(1) line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR. 1 Introduction Maximum Independent Set of Rectangles (MISR) is a fundamental and well-studied problem in computational geometry, combinatorial optimization and approximation algorithms. In MISR, we are given a set R of n possibly overlapping axis- parallel rectangles in the plane. We are looking for a subset R ∗ ⊆R of maximum cardinality such that the rectangles in R ∗ are pairwise disjoint. MISR finds numerous applications in practice, e.g., in map labeling [37, 24], data mining [27] and resource allocation [44]. The problem is an important special case of the MAXIMUM I NDEPENDENT SET problem in graphs, which in general is NP-hard to approximate within a factor of n 1−ε for any constant ε> 0 [36]. However, for MISR much better approximation ratios are possible, e.g., there are multiple O(log n)-approximation algorithms [15, 51, 43]. It had been a long-standing open problem to find an O(1)-approximation algorithm for MISR. One possible approach for this is to compute an optimal solution to the canonical LP-relaxation of MISR and round it. This approach was used by Chalermsook and Chuzhoy in order to obtain an O(log log n)-approximation [16]. The LP is conjectured to have an integrality gap of O(1) which is a long-standing open problem by itself, with interesting connections to graph theory [14, 16]. On the other hand, it seems likely that MISR admits even a PTAS, given that it admits a QPTAS due to Adamaszek and Wiese [3], and in particular one with a running time of only n O((log log n/ε) 4 ) due to Chuzhoy and Ene [22]. Recently, in a breakthrough result, Mitchell presented a polynomial time 10-approximation algorithm [46] and consequently solved the aforementioned long-standing open problem. Instead of rounding the LP, he employs a recursive partitioning of the plane into a special type of rectilinear polygons called corner-clipped rectangles (CCRs). Given a CCR, * Universidad de O’Higgins, Chile. waldo.galvez@uoh.cl This project was carried out when the author was a postdoctoral researcher at the Technical University of Munich in Germany. Supported by the European Research Council, Grant Agreement No. 691672, project APEG. † Indian Institute of Science, Bengaluru, India. arindamkhan@iisc.ac.in Supported by Pratisksha Trust Young Investigator Award. ‡ University of Warsaw and IDEAS NCBR, Poland. mathieu.mari@ens.fr Partially supported by the ERC CoG grant TUgbOAT no 772346. § University of Augsburg, Germany. moemke@informatik.uni-augsburg.de Partially supported by the DFG Grant 439522729 (Heisenberg- Grant) and DFG Grant 439637648 (Sachbeihilfe). ¶ Indian Institute of Technology, Kharagpur, India. pmsreddifeb18@gmail.com || Universidad de Chile, Chile. awiese@dii.uchile.cl Partially supported by FONDECYT Regular grant 1200173. Copyright © 2022 by SIAM Unauthorized reproduction of this article is prohibited 894 Downloaded 01/08/22 to 54.91.149.201 Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms