# Approximate Guarding of Monotone and Rectilinear Polygons

@article{Nilsson2005ApproximateGO,
title={Approximate Guarding of Monotone and Rectilinear Polygons},
author={Bengt J. Nilsson},
journal={Algorithmica},
year={2005},
volume={66},
pages={564-594}
}
We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT2).
6 Citations

### On Orthogonally Guarding Orthogonal Polygons with Bounded Treewidth

• Computer Science, Mathematics
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• 2020
This paper shows that the problem of finding the minimum number of guards in all these guarding models becomes linear-time solvable on polygons with bounded treewidth.

### Exact Algorithms for Terrain Guarding

• Mathematics
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• 2017
An nO(√ k)-time algorithm is developed and it is shown that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guards, is fixed-parameter tractable and admits a subexponential-time algorithm.

### An Approximation Algorithm for the Art Gallery Problem

• Mathematics
SoCG
• 2017
The first $O(\log \text{OPT})$-approximation algorithm for the point guard problem for simple polygons is presented and a mistake is pointed out in the latter.

### On Guarding Orthogonal Polygons with Sliding Cameras

• Mathematics, Computer Science
WALCOM
• 2017
This paper gives the first constant-factor approximation algorithm for the problem of guarding P with the minimum number of sliding cameras and shows that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth.

### Parameterized Hardness of Art Gallery Problems

• Mathematics
ESA
• 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 Hardness of the Art Gallery Problem

• Mathematics
ArXiv
• 2016
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.

## References

SHOWING 1-10 OF 40 REFERENCES

### Approximate Guarding of Monotone and Rectilinear Polygons

• Computer Science, Mathematics
Algorithmica
• 2012
It is shown that vertex guarding amonotone polygon is NP-hard and a constant factor approximation algorithm for interior guarding monotone polygons is constructed that has an approximation factor independent of the number of vertices of the polygon.

### Recent Results in Art Galleries

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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