# Approximability of guarding weak visibility polygons

@article{Bhattacharya2017ApproximabilityOG, title={Approximability of guarding weak visibility polygons}, author={Pritam Bhattacharya and Subir Kumar Ghosh and Bodhayan Roy}, journal={Discret. Appl. Math.}, year={2017}, volume={228}, pages={109-129} }

## 13 Citations

### 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.

### 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.

### Implementation of polygon guarding algorithms for art gallery problems

- Computer ScienceArXiv
- 2022

Experiments show that this algorithm assigns near optimal guards for guarding the input polygons, and is provided as a new algorithm that uses Ghosh’s idea.

### On conflict-free chromatic guarding of simple polygons

- Mathematics, Computer ScienceCOCOA
- 2019

The problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour is studied, and an upper bound of O(log^2 n) colours on n-vertex weak visibility polygons is given, which generalises to all simple polygons.

### Guarding Weakly-Visible Polygons with Half-Guards

- Mathematics, Computer Science
- 2022

A polynomial time approximation scheme for vertex guarding a weakly-visible polygon with half-guards is given and it is shown that even with many restrictions, the problem is NP-hardness.

### On the Complexity of Half-Guarding Monotone Polygons

- Computer Science, MathematicsLATIN
- 2022

A variant of the art gallery problem where all guards are limited to seeing to the right inside a monotone polygon with half-guards is considered, which provides a polynomial-time approximation for point guarding the entire monot one polygon and provides an NP-hardness reduction.

### Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains

- MathematicsWAOA
- 2019

A local-search-based PTAS for minimum dominating set in terrain-like graphs and non-jumping graphs is presented, and it is observed that both families admit PTASs for maximum independent set.

### Vertex guarding for dynamic orthogonal art galleries

- MathematicsInt. J. Comput. Geom. Appl.
- 2021

A dynamic algorithm for guarding with vertex guards is devised, i.e., whenever orthogonal polygon is modified, algorithm updates the set of vertex guards and their positions for guarding the modified orthogonic polygon.

### A PTAS for vertex guarding weakly-visible polygons - An extended abstract

- MathematicsArXiv
- 2018

This extended abstract presents a PTAS for guarding the vertices of a weakly-visible polygon from a subset of its vertices, and shows how to obtain aPTAS for vertex guarding the entire polygon.

## References

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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.

### Improved Approximation for Guarding Simple Galleries from the Perimeter

- Computer ScienceDiscret. Comput. Geom.
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This algorithm is the first to improve upon O(log opt)-approximation algorithms that use generic net finders for set systems of finite VC-dimension.

### Inapproximability Results for Guarding Polygons and Terrains

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- 2001

This paper proves that if the input polygon has no holes, there is a constant δ >0 such that no polynomial time algorithm can achieve an approximation ratio of 1+δ, for each of these guard problems, and shows inapproximability for the POINT GUARD problem for polygons with holes.

### Inapproximability of some art gallery problems

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We prove that the three art gallery problems Vertex Guard Edge Guard and Point Guard for simple polygons with holes cannot be approximated by any polynomial time algorithm with a ratio of lnn for…

### Computational complexity of art gallery problems

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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.

### A Pseudopolynomial Time O (log n )-Approximation Algorithm for Art Gallery Problems

- Computer Science, MathematicsWADS
- 2007

A O(log copt)-approximation algorithm for the point guard problem where copt is the optimal number of guards and it is found that the optimal solution to the art gallery problem where guards are restricted to this set is at most 3copt.

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- Computer ScienceWADS
- 1991

It is proved that finding the minimum number of vision points along a shortest watchman route is NP-hard, and the notion of vision spans along a path (route) which provide a natural connection between the (stationary) Art Gallery problem, the m-watchmen problem and the watch man route problem is introduced.

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- Computer Science, MathematicsDiscret. Appl. Math.
- 2010