# Inapproximability Results for Guarding Polygons and Terrains

@article{Eidenbenz2001InapproximabilityRF, title={Inapproximability Results for Guarding Polygons and Terrains}, author={Stephan Johannes Eidenbenz and Christoph Stamm and Peter Widmayer}, journal={Algorithmica}, year={2001}, volume={31}, pages={79-113} }

Abstract. Past research on art gallery problems has concentrated almost exclusively on bounds on the numbers of guards needed in the worst case in various settings. On the complexity side, fewer results are available. For instance, it has long been known that placing a smallest number of guards for a given input polygon is NP -hard. In this paper we initiate the study of the approximability of several types of art gallery problems. Motivated by a practical problem, we study the approximation…

## 142 Citations

### Optimum Inapproximability Results for Finding Minimum Hidden Guard Sets in Polygons and Terrains

- Mathematics, Computer ScienceSWAT
- 2002

We study the problem Minimum Hidden Guard Set, which consists of positioning a minimum number of guards in a given polygon or terrain such that no two guards see each other and such that every point…

### Finding minimum hidden guard sets in polygons - tight approximability results

- Computer Science, MathematicsComput. Geom.
- 2006

### Exact solutions and bounds for general art gallery problems

- Computer Science, MathematicsJEAL
- 2012

A primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and—in case of convergence and integrality—ends with an optimal solution to the classical Art Gallery Problem.

### Approximability of guarding weak visibility polygons

- Computer Science, MathematicsDiscret. Appl. Math.
- 2017

### Vertex Guarding in Weak Visibility Polygons

- Computer ScienceCALDAM
- 2015

The conjecture that constant-factor approximation algorithms exist for the special class of polygons that are weakly visible from an edge and contain no holes is settled by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in \(\mathcal{O}(n^2)\) time.

### Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards

- Computer Science, MathematicsArXiv
- 2017

Three polynomial-time algorithms with a constant approximation ratio for guarding an $n$-sided simple polygon $P$ using vertex guards are presented, settling the conjecture by Ghosh regarding the existence of constant-factor approximation algorithms for this problem.

### Guarding problems and geometric split trees

- Computer Science, Mathematics
- 2011

A significant open problem is resolved by proving strong NP-completeness for terrain guarding and an approximation algorithm for guarding simple polygons with perimeter guards is presented, and a random tree model is introduced that is general while still allowing powerful comparisons with random trees from more restricted models.

### The Parameterized Complexity of Guarding Almost Convex Polygons

- MathematicsSoCG
- 2020

Structural properties of "almost convex polygons" are utilized to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

### Irrational Guards are Sometimes Needed∗

- Mathematics
- 2017

In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so…

### A PTAS for the Continuous 1.5D Terrain Guarding Problem

- MathematicsCCCG
- 2014

It is shown that for the general problem, the continuous 1.5-dimensional terrain guarding problem, one can construct nite guard and witness sets, G and W, such that there exists an optimal guard cover G G that covers T, and when these guards monitor all points in W the entire terrain is guarded.

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