Polygon triangulation: efficiency and minimality
Journal of Algorithms
Worst-case optimal algorithms for constructing visibility polygons with holes
SCG '86 Proceedings of the second annual symposium on Computational geometry
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Journal of Algorithms
The Power of Non-Rectilinear Holes
Proceedings of the 9th Colloquium on Automata, Languages and Programming
An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee
SIAM Journal on Computing
Maximum Matchings via Gaussian Elimination
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Minimum weight triangulation is NP-hard
Proceedings of the twenty-second annual symposium on Computational geometry
Note: Approximation algorithms for art gallery problems in polygons
Discrete Applied Mathematics
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We consider the triangle cover problem. Given a polygon P, cover it with a minimum number of triangles contained in P. This is a generalization of the well-known polygon triangulation problem. Another way to look at it is as a restriction of the convex cover problem, in which a polygon has to be covered with a minimum number of convex pieces. Answering a question stated in the Handbook of Discrete and Computational Geometry, we show that the convex cover problem without Steiner points is NP-hard. We present a reduction that also implies NP-hardness of the triangle cover problem and which in a second step allows to get rid of Steiner points. For the problem where only the boundary of the polygon has to be covered, we also show that it is contained in NP and thus NP-complete and give an efficient factor 2 approximation algorithm.