@article{6b87e515234c4fd79cece701463fed5b,

title = "Approximation schemes for independent set and sparse subsets of polygons",

abstract = "We present a (1 + ε)-approximation algorithm with quasi-polynomial running time for computing a maximum weight independent set of polygons from a given set of polygons in the plane. Contrasting this, the best-known polynomial time algorithm for the problem has an approximation ratio of nε. Surprisingly, we can extend the algorithm to the problem of computing the maximum cardinality subset of the given set of polygons whose intersection graph fulfills some sparsity condition. For example, we show that one can approximate the maximum subset of polygons such that the intersection graph of the subset is planar or does not contain a cycle of length 4 (i.e., K2,2). Our algorithm relies on a recursive partitioning scheme, whose backbone is the existence of balanced cuts with small complexity that intersect polygons from the optimal solution of a small total weight. For the case of large axis-parallel rectangles, we provide a polynomial time (1 + ε)-approximation for the maximum weight independent set. Specifically, we consider the problem where each rectangle has one edge whose length is at least a constant fraction of the length of the corresponding edge of the bounding box of all the input elements. This is now the most general case for which a PTAS is known, and it requires a new and involved partitioning scheme, which should be of independent interest.",

keywords = "Approximation algorithms, Approximation schemes, Independent set, Rectangles",

author = "Anna Adamaszek and Sariel Har-Peled and Andreas Wiese",

note = "Funding Information: Anna Adamaszek supported by the Danish Council for Independent Research DFF-MOBILEX mobility grant. Work on this article Sariel Har-Peled was partially supported by a NSF AF award CCF-1217462. Authors{\textquoteright} addresses: A. Adamaszek, Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark; email: anna@mpi-inf.mpg.de; S. Har-Peled, 201 N. Goodwin Avenue, Urbana, IL 61801-2302, USA; email: sariel@illinois.edu; A. Wiese (corresponding author), Departamento de Ingenier{\'i}a Industrial, Facultad de Ciencias F{\'i}sicas y Matem{\'a}ticas, Universidad de Chile, Beauchef 851 Of. 705 Piso 7, Santiago Centro, Chile; email: awiese@dii.uchile.cl. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. {\textcopyright} 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. 0004-5411/2019/06-ART29 $15.00 https://doi.org/10.1145/3326122 Funding Information: Anna Adamaszek supported by the Danish Council for Independent Research DFF-MOBILEX mobility grant. Work on this article Sariel Har-Peled was partially supported by a NSF AF award CCF-1217462. Publisher Copyright: {\textcopyright} 2019 Copyright held by the owner/author(s).",

year = "2019",

month = jun,

day = "17",

doi = "10.1145/3326122",

language = "English (US)",

volume = "66",

journal = "Journal of the ACM",

issn = "0004-5411",

publisher = "Association for Computing Machinery (ACM)",

number = "4",

}