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- W4254726953 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 $frac{1}{2}+ frac{2 +sqrt{2}}{pi}=1.5867ldots$ . For connected barriers we achieve the approximation ratio 1.5716, while for single-arc barriers we achieve the approximation ratio $frac{pi+5}{pi+2} = 1.5834ldots$ . 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." @default.
- W4254726953 created "2022-05-12" @default.
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- W4254726953 date "2012-12-28" @default.
- W4254726953 modified "2023-10-06" @default.
- W4254726953 title "Opaque Sets" @default.
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- W4254726953 doi "https://doi.org/10.1007/s00453-012-9735-2" @default.
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