Beyond Triangulation: Covering Polygons with Triangles
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.
KeywordsCover Problem Steiner Point Simple Polygon Polygonal Region Main Switch
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