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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.128
URN: urn:nbn:de:0030-drops-46938
URL: http://drops.dagstuhl.de/opus/volltexte/2014/4693/
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Dumitrescu, Adrian ; Jiang, Minghui ; Tóth, Csaba D.

Computing Opaque Interior Barriers à la Shermer

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Abstract

The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given polygon was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the polygon. In 1991 Shermer proposed an exponential-time algorithm that computes an interior-restricted barrier made of segments for any given convex n-gon. He conjectured that the barrier found by his algorithm is optimal, however this was refuted recently by Provan et al. Here we give a Shermer like algorithm that computes an interior polygonal barrier whose length is at most 1.7168 times the optimal and that runs in O(n) time. As a byproduct, we also deduce upper and lower bounds on the approximation ratio of Shermer's algorithm.

BibTeX - Entry

@InProceedings{dumitrescu_et_al:LIPIcs:2014:4693,
  author =	{Adrian Dumitrescu and Minghui Jiang and Csaba D. T{\'o}th},
  title =	{{Computing Opaque Interior Barriers {\`a} la Shermer}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{128--143},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4693},
  URN =		{urn:nbn:de:0030-drops-46938},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.128},
  annote =	{Keywords: Opaque barrier, approximation algorithm, isoperimetric inequality}
}

Keywords: Opaque barrier, approximation algorithm, isoperimetric inequality
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Issue Date: 2014
Date of publication: 02.09.2014


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