# The art gallery problem is ∃ ℝ-complete

@article{Abrahamsen2018TheAG, title={The art gallery problem is ∃ ℝ-complete}, author={Mikkel Abrahamsen and Anna Adamaszek and Tillmann Miltzow}, journal={Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing}, year={2018} }

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the…

## 67 Citations

### Complexity of the Boundary-Guarding Art Gallery Problem

- Mathematics
- 2022

We resolve the complexity of the boundary-guarding variant of the art gallery problem, showing that it is ∃ R -complete, meaning that it is equivalent under polynomial time reductions to deciding…

### 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.

### 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.

### The Dispersive Art Gallery Problem

- Computer Science, MathematicsArXiv
- 2022

It is proved that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP -complete.

### 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.

### 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.

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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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The problem of determining the minimum number of vertex guards that can see an n -wall simply connected art gallery is shown to be NP-hard, and the proof can be modified to show that the problems ofetermining theminimum number of edge guards and theMinimum number of point guards in a simply connected polygonal region are also NP- hard.

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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.

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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…

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