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Computing Opaque Interior Barriers à la Shermer

The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given planar convex body was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the convex body. 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, but 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.

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