The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given polygon was initiated by#R##N#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.
['Adrian Dumitrescu', 'Minghui Jiang', 'Csaba D. Tóth']
Computing Opaque Interior Barriers à la Shermer