License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2020.35
URN: urn:nbn:de:0030-drops-122821
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12282/
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Spoerhase, Joachim ; Storandt, Sabine ; Zink, Johannes

Simplification of Polyline Bundles

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LIPIcs-SWAT-2020-35.pdf (1 MB)

Abstract

We propose and study a generalization to the well-known problem of polyline simplification. Instead of a single polyline, we are given a set of l polylines possibly sharing some line segments and bend points. Our goal is to minimize the number of bend points in the simplified bundle with respect to some error tolerance δ (measuring Fréchet distance) but under the additional constraint that shared parts have to be simplified consistently. We show that polyline bundle simplification is NP-hard to approximate within a factor n^(1/3 - ε) for any ε > 0 where n is the number of bend points in the polyline bundle. This inapproximability even applies to instances with only l=2 polylines. However, we identify the sensitivity of the solution to the choice of δ as a reason for this strong inapproximability. In particular, we prove that if we allow δ to be exceeded by a factor of 2 in our solution, we can find a simplified polyline bundle with no more than ?(log (l + n)) ⋅ OPT bend points in polytime, providing us with an efficient bi-criteria approximation. As a further result, we show fixed-parameter tractability in the number of shared bend points.

BibTeX - Entry

@InProceedings{spoerhase_et_al:LIPIcs:2020:12282,
  author =	{Joachim Spoerhase and Sabine Storandt and Johannes Zink},
  title =	{{Simplification of Polyline Bundles}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Susanne Albers},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12282},
  URN =		{urn:nbn:de:0030-drops-122821},
  doi =		{10.4230/LIPIcs.SWAT.2020.35},
  annote =	{Keywords: Polyline Simplification, Bi-criteria Approximation, Hardness of Approximation, Geometric Set Cover}
}

Keywords: Polyline Simplification, Bi-criteria Approximation, Hardness of Approximation, Geometric Set Cover
Collection: 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)
Issue Date: 2020
Date of publication: 12.06.2020

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