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Simplification of Polyline Bundles

Authors Joachim Spoerhase , Sabine Storandt, Johannes Zink

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Fixed parameter tractability
Keywords
  • Polyline Simplification
  • Bi-criteria Approximation
  • Hardness of Approximation
  • Geometric Set Cover

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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 𝒪(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.

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Joachim Spoerhase, Sabine Storandt, and Johannes Zink. Simplification of Polyline Bundles. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.35

Author Details

Joachim Spoerhase
  • Aalto University, Finland
  • University of Würzburg, Germany
Sabine Storandt
  • University of Konstanz, Germany
Johannes Zink
  • University of Würzburg, Germany

Funding

  • Spoerhase, Joachim: Funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 759557).
  • Storandt, Sabine: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 50974019 – TRR 161.

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