License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2017.42
URN: urn:nbn:de:0030-drops-80917
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8091/
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Pilipczuk, Michal ; van Leeuwen, Erik Jan ; Wiese, Andreas

Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking

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Abstract

Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the approximation and in the parameterized setting. The best known polynomial-time approximation algorithms achieve super-constant approximation ratios [Chalermsook & Chuzhoy, Proc. SODA 2009; Chan & Har-Peled, Discrete & Comp. Geometry, 2012], even though there is a (1+epsilon)-approximation running in quasi-polynomial time [Adamaszek & Wiese, Proc. FOCS 2013; Chuzhoy & Ene, Proc. FOCS 2016]. When parameterized by the target size of the solution, the problem is W[1]-hard even in the unweighted setting [Marx, ESA 2005].

To achieve tractability, we study the following shrinking model: one is allowed to shrink each input rectangle by a multiplicative factor 1-delta for some fixed delta > 0, but the performance is still compared against the optimal solution for the original, non-shrunk instance. We prove that in this regime, the problem admits an EPTAS with running time f(epsilon,delta) n^{O(1)}, and an FPT algorithm with running time f(k,delta) n^{O(1)}, in the setting where a maximum-weight solution of size at most k is to be computed. This improves and significantly simplifies a PTAS given earlier for this problem [Adamaszek, Chalermsook & Wiese, Proc. APPROX/RANDOM 2015], and provides the first parameterized results for the shrinking model. Furthermore, we explore kernelization in the shrinking model, by giving efficient kernelization procedures for several variants of the problem when the input rectangles are squares.

BibTeX - Entry

@InProceedings{pilipczuk_et_al:LIPIcs:2017:8091,
  author =	{Michal Pilipczuk and Erik Jan van Leeuwen and Andreas Wiese},
  title =	{{Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8091},
  URN =		{urn:nbn:de:0030-drops-80917},
  doi =		{10.4230/LIPIcs.MFCS.2017.42},
  annote =	{Keywords: Combinatorial optimization, Approximation algorithms, Fixed-parameter algorithms}
}

Keywords: Combinatorial optimization, Approximation algorithms, Fixed-parameter algorithms
Collection: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Issue Date: 2017
Date of publication: 01.12.2017

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