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Algorithmica

, Volume 69, Issue 2, pp 315–334 | Cite as

Opaque Sets

  • Adrian Dumitrescu
  • Minghui JiangEmail author
  • János Pach
Article

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.

Keywords

Opaque set Opaque polygon problem Point goalie problem Traveling salesman problem Approximation algorithm Cauchy’s surface area formula 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA
  3. 3.Ecole Polytechnique Fédérale de Lausanne and City CollegeNew YorkUSA

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