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Opaque Sets

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Abstract

The problem of finding “small” sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an opaque set or a barrier for that region. We consider the problem of computing the shortest barrier for a given convex polygon with n vertices. No exact algorithm is currently known even for the simplest instances such as a square or an equilateral triangle. For general barriers, we present an approximation algorithm with ratio \(\frac{1}{2}+ \frac{2 +\sqrt{2}}{\pi}=1.5867\ldots\) . For connected barriers we achieve the approximation ratio 1.5716, while for single-arc barriers we achieve the approximation ratio \(\frac{\pi+5}{\pi+2} = 1.5834\ldots\) . All three algorithms run in O(n) time. We also show that if the barrier is restricted to the (interior and the boundary of the) input polygon, then the problem admits a fully polynomial-time approximation scheme for the connected case and a quadratic-time exact algorithm for the single-arc case.

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, USA

    Adrian Dumitrescu

  2. Department of Computer Science, Utah State University, Logan, USA

    Minghui Jiang

  3. Ecole Polytechnique Fédérale de Lausanne and City College, New York, USA

    János Pach

Authors
  1. Adrian Dumitrescu
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  2. Minghui Jiang
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  3. János Pach
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Corresponding author

Correspondence to Minghui Jiang.

Additional information

A. Dumitrescu was supported in part by NSF CAREER grant CCF-0444188 and by NSF grant DMS-1001667. Part of the research by this author was done at Ecole Polytechnique Fédérale de Lausanne.

M. Jiang was supported in part by NSF grant DBI-0743670.

J. Pach research was partially supported by NSF grant CCF-08-30272, grants from OTKA, SNF, and PSC-CUNY.

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Dumitrescu, A., Jiang, M. & Pach, J. Opaque Sets. Algorithmica 69, 315–334 (2014). https://doi.org/10.1007/s00453-012-9735-2

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