# Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards

The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a polygon P, is fully guarded. Most standard versions of this problem are known to be NP-hard. In 1987, Ghosh provided a deterministic O( n)-approximation algorithm for the case of vertex guards and edge guards in simple polygons. In the same paper, Ghosh also conjectured the existence of constant ratio approximation algorithms for these problems. We present here three polynomial-time algorithms with a constant approximation ratio for guarding an n-sided simple polygon P using vertex guards. (i) The first algorithm, that has an approximation ratio of 18, guards all vertices of P in O(n^4) time. (ii) The second algorithm, that has the same approximation ratio of 18, guards the entire boundary of P in O(n^5) time. (iii) The third algorithm, that has an approximation ratio of 27, guards all interior and boundary points of P in O(n^5) time. The significance of our results lies in the fact that these results settle the conjecture by Ghosh regarding the existence of constant-factor approximation algorithms for this problem, which has been open since 1987 despite several attempts by researchers. Our approximation algorithms exploit several deep visibility structures of simple polygons which are interesting in their own right.

• 1 publication
• 4 publications
• 1 publication
research
07/02/2019

### A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon

The problem of vertex guarding a simple polygon was first studied by Sub...
research
09/26/2019

### Approximation Algorithms for Process Systems Engineering

Designing and analyzing algorithms with provable performance guarantees ...
research
09/22/2018

### A 2-Approximation Algorithm for Feedback Vertex Set in Tournaments

A tournament is a directed graph T such that every pair of vertices is ...
research
06/25/2019

research
07/10/2022

### Opposing Half Guards

We study the art gallery problem for opposing half guards: guards that c...
research
09/16/2018

### Constant factor FPT approximation for capacitated k-median

Capacitated k-median is one of the few outstanding optimization problems...
research
09/08/2019

### Target Set Selection for Conservative Populations

Let G = (V,E) be a graph on n vertices, where d_v denotes the degree of ...