Computational geometry: an introduction
Computational geometry: an introduction
Computational complexity of art gallery problems
IEEE Transactions on Information Theory
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Learnability and the Vapnik-Chervonenkis dimension
Journal of the ACM (JACM)
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
The Robot Localization Problem
SIAM Journal on Computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Approximation algorithms
Lectures on Discrete Geometry
Inapproximability Results for Guarding Polygons without Holes
ISAAC '98 Proceedings of the 9th International Symposium on Algorithms and Computation
Efficient visibility queries in simple polygons
Computational Geometry: Theory and Applications
The art gallery theorem: its variations, applications and algorithmic aspects
The art gallery theorem: its variations, applications and algorithmic aspects
Algorithmic construction of sets for k-restrictions
ACM Transactions on Algorithms (TALG)
Guarding galleries and terrains
Information Processing Letters
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Hitting sets when the VC-dimension is small
Information Processing Letters
Note: Approximation algorithms for art gallery problems in polygons
Discrete Applied Mathematics
Improved Approximation for Guarding Simple Galleries from the Perimeter
Discrete & Computational Geometry
Approximation algorithms for art gallery problems in polygons and terrains
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Guarding problems and geometric split trees
Guarding problems and geometric split trees
Some NP-hard polygon decomposition problems
IEEE Transactions on Information Theory
A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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For a polygon P with n vertices, the vertex guarding problem asks for the minimum subset G of P@?s vertices such that every point in P is seen by at least one point in G. This problem is NP-complete and APX-hard. The first approximation algorithm (Ghosh, 1987) involves decomposing P into O(n^4) cells that are equivalence classes for visibility from the vertices of P. This discretized problem can then be treated as an instance of Set Cover and solved in O(n^5) time with a greedy O(logn)-approximation algorithm. Ghosh (2010) recently revisited the algorithm, noting that minimum visibility decompositions for simple polygons (Bose et al., 2000) have only O(n^3) cells, improving the running time of the algorithm to O(n^4) for simple polygons. In this paper we observe that, since minimum visibility decompositions for simple polygons have only O(n^2) sinks (i.e., cells of minimal visibility) (Bose et al., 2000), the running time of the algorithm can be further improved to O(n^3). We then extend the result of Bose et al. to polygons with holes, showing that a minimum visibility decomposition of a polygon with h holes has only O((h+1)n^3) cells and only O((h+1)^2n^2) cells of minimal visibility. We exploit this result to obtain a faster algorithm for vertex guarding polygons with holes. We then show that, in the same time complexity, we can attain approximation factors of O(loglogopt) for simple polygons and O((1+log(h+1))logopt) for polygons with holes.