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 a O(n) time approximation algorithm with ratio \(\frac{1}{2}+ \frac{2 +\sqrt{2}}{\pi}=1.5867\ldots\). For connected barriers, we can achieve the approximation ratio \(\frac{\pi+5}{\pi+2} =1.5834\ldots\) again 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. These are the first approximation algorithms obtained for this problem.
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Dumitrescu, A., Jiang, M., Pach, J. (2011). Opaque Sets. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_17
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