Opaque sets

TL;DR

This paper studies the problem of finding the shortest barrier that intersects all lines crossing a convex polygon, providing approximation algorithms with specific ratios and efficient algorithms for restricted cases.

Contribution

It introduces approximation algorithms with provable ratios for the shortest barrier problem for convex polygons and offers efficient algorithms for restricted barrier types.

Findings

Approximation ratio of 1.5867% for general barriers.

Approximation ratio of 1.5716% for connected barriers.

Quadratic-time exact algorithm for barriers restricted to the polygon interior and boundary.

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 {\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.