Opaque sets

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 {\em opaque set} or a {\em 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 $1/2 + \frac{2 +\sqrt{2}}{\pi}=1.5867…$. For connected barriers we achieve the approximation ratio 1.5716, while for single-arc barriers we achieve the approximation ratio $\frac{\pi+5}{\pi+2} = 1.5834…$. 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.

💡 Research Summary

The paper revisits the classic “opaque set” (or barrier) problem originally posed by Mazurkiewicz in 1916: given a convex region C in the plane, find a set B of minimum total length that intersects every straight line intersecting C. While the problem is trivial for a line segment (the segment itself), it becomes highly non‑trivial for even the simplest convex shapes such as a unit square or an equilateral triangle, and no exact algorithm is known for these cases.

The authors focus on convex polygons P with n vertices and consider three natural families of barriers: (i) arbitrary (possibly disconnected) barriers, (ii) connected barriers, and (iii) single‑arc (continuous) barriers. For each family they design a linear‑time (O(n)) approximation algorithm with a provably bounded ratio to the optimal solution within that family. The main results are:

• Arbitrary barriers – an algorithm achieving a ratio
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