Computing Opaque Interior Barriers à la Shermer

Adrian Dumitrescu,Csaba D Tóth,Minghui Jiang

doi:10.1137/14098805x

Abstract

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.

Highlights

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916 [21]; see [2, 14]
What is the length of the shortest barrier for a given convex body B? In spite of considerable efforts, the answer to this question is not known even in the simplest instances, such as when B is a square, a disk, or an equilateral triangle; see [3], [4, Problem A30], [10], [11], [12], [13, Section 8.11], [15, Problem 12]
When B is a unit square, the barrier in Figure 1(right) is conjectured to be optimal; on the other hand, the current best lower bound on the length of a barrier was only 2 until very recently; the earliest record for this bound of 2 dates back to Jones in 1964 [16]

Summary

Introduction

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916 [21]; see [2, 14]. We present the first nontrivial approximation algorithm with ratio below 2 for computing an interior barrier for a given convex polygon. Shermer conjectured that a shortest interior barrier (he calls it an “opaque forest”) of a given convex polygon P with n vertices can be generated by an instance of the following procedure:. There exist convex polygons (e.g., a rhombus) for which Shermer’s algorithm returns an interior barrier that is at least 1.00769 times longer than the optimal. It is well-known that the number of triangulations of a convex n-gon is Cn−2, where. The approximation ratio of Shermer’s procedure is at most 1.7168 and at least 1.00769

Preliminaries

Proof of Theorem 1

A Lower Bound on the Approximation Ratio of Shermer’s Algorithm

Concluding Remarks