APX-Hardness of the Minimum Vision Points Problem

TL;DR

This paper proves that determining the minimum number of vision points needed for a watchman route in polygons is APX-Hard, highlighting computational complexity challenges in this geometric optimization problem.

Contribution

It establishes the APX-Hardness of the minimum vision points problem for simple and certain rectilinear polygons, advancing understanding of its computational difficulty.

Findings

Proves APX-Hardness for simple polygons

Extends proof to rectilinear polygons with up to three dent orientations

Highlights complexity of optimizing watchman routes

Abstract

Placing a minimum number of guards on a given watchman route in a polygonal domain is called the {\em minimum vision points problem}. We prove that finding the minimum number of vision points on a shortest watchman route in a simple polygon is APX-Hard. We then extend the proof to the class of rectilinear polygons having at most three dent orientations.