On approximations to minimum link visibility paths in simple polygons

TL;DR

This paper presents an approximation algorithm for the NP-hard problem of finding minimum link visibility paths in simple polygons, achieving near-optimal solutions with a runtime dependent on a parameter k.

Contribution

It introduces a new approximation algorithm with a runtime of O(kn^2) that bounds the link-length within a factor related to the optimal, using a novel combination of existing approximation techniques.

Findings

Provides an algorithm with approximation factor depending on parameter k

Achieves near-optimal link-length with polynomial runtime

Extends approximation methods to visibility path problems in polygons

Abstract

We investigate a practical variant of the well-known polygonal visibility path (watchman) problem. For a polygon $P$, a minimum link visibility path is a polygonal visibility path in $P$ that has the minimum number of links. The problem of finding a minimum link visibility path is NP-hard for simple polygons. If the link-length (number of links) of a minimum link visibility path (tour) is $Opt$ for a simple polygon $P$ with $n$ vertices, we provide an algorithm with $O(kn^2)$ runtime that produces polygonal visibility paths (or tours) of link-length at most $(\gamma+a_l/(k-1))Opt$ (or $(\gamma+a_l/k)Opt$), where $k$ is a parameter dependent on $P$, $a_l$ is an output sensitive parameter and $\gamma$ is the approximation factor of an $O(k^3)$ time approximation algorithm for the graphic traveling salesman problem (path or tour version).