# An Approximation Algorithm for the Art Gallery Problem

@inproceedings{Bonnet2017AnAA, title={An Approximation Algorithm for the Art Gallery Problem}, author={{\'E}douard Bonnet and Tillmann Miltzow}, booktitle={International Symposium on Computational Geometry}, year={2017} }

Given a simple polygon $\mathcal{P}$ on $n$ vertices, two points $x,y$ in $\mathcal{P}$ are said to be visible to each other if the line segment between $x$ and $y$ is contained in $\mathcal{P}$. The Point Guard Art Gallery problem asks for a minimum set $S$ such that every point in $\mathcal{P}$ is visible from a point in $S$. The set $S$ is referred to as guards. Assuming integer coordinates and a specific general position assumption, we present the first $O(\log \text{OPT})$-approximation…

## 27 Citations

### A Practical Algorithm with Performance Guarantees for the Art~Gallery Problem

- Computer ScienceSoCG
- 2021

A one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program guarantees to find the optimal solution for every vision stable polygon.

### Smoothed Analysis of the Art Gallery Problem

- Computer ScienceArXiv
- 2018

The results are that algebraic methods are not needed to solve the Art Gallery Problem in typical instances and the expected number of bits to describe optimal guard positions per guard is logarithmic in the input and the magnitude of the perturbation.

### 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.

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

- Mathematics, Computer ScienceWAOA
- 2020

This paper presents a $(2+\varepsilon)-approximation algorithm for guarding a weakly visible polygon, and presents two algorithms based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygon's boundary.

### Parameterized Complexity of Geometric Covering Problems Having Conflicts

- Mathematics, Computer ScienceAlgorithmica
- 2019

It is proved that conflict-free version of Covering Points by Intervals does not admit an FPT algorithm, unless FPT =W[1], for the family of conflict graphs for which the Independent Set problem is W[1]-hard.

### A Constant-Factor Approximation Algorithm for Point Guarding an Art Gallery

- Computer ScienceArXiv
- 2021

This paper proposes an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid and the running time depends on the number of cells of the grid.

### Parameterized Hardness of Art Gallery Problems

- MathematicsESA
- 2016

Lower bounds almost match the n^{O(k)} algorithms that exist for both problems, and rule out a f(k)*n^{o(k/log k)} algorithm for any computable function f, where k := |S| is the number of guards.

### Irrational Guards are Sometimes Needed

- MathematicsSoCG
- 2017

It is shown that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution of the art gallery problem.

### Topological Art in Simple Galleries

- MathematicsSOSA
- 2022

It is shown that for every semi-algebraic set S there is a polygon P such that V (P ) is homotopy equivalent to S, and that for various concrete topological spaces T, instances I of the art gallery problem such as V (I) is homeomorphic to T.

### Orthogonal Terrain Guarding is NP-complete

- Mathematics, Computer ScienceSoCG
- 2018

This paper adapts the gadgets of King and Krohn to rectilinear terrains to prove that even Orthogonal Terrain Guarding is NP-complete, and shows how the reduction from Planar 3-SAT can actually be made linear (instead of quadratic).

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Lower bounds almost match the $n^{O(k)}$ algorithms that exist for both problems, and rule out any $f(k)n^{o(k / \log k)}$ algorithm, for any computable function $f$, unless the Exponential Time Hypothesis fails.

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Lower bounds almost match the n^{O(k)} algorithms that exist for both problems, and rule out a f(k)*n^{o(k/log k)} algorithm for any computable function f, where k := |S| is the number of guards.

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Two points in a polygon are called visible if the straight line segment between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in…