Triangulating a nonconvex polytope
Discrete & Computational Geometry - Selected papers from the fifth annual ACM symposium on computational geometry, Saarbrücken, Germany, June 5-11, 1989
Tetrahedrizing point sets in three dimensions
Journal of Symbolic Computation
Subdivisions and triangulations of polytopes
Handbook of discrete and computational geometry
Finding minimal triangulations of convex 3-polytopes is NP-hard
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Approximation for minimum triangulation of convex polyhedra
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
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Finding minimum triangulations of convex polyhedra is NP-hard. The best approximation algorithms only give a ratio 2 for this problem, and for combinatorial algorithms it is shown to be the best possible asymptotically. In this paper we improve the approximation ratio of finding minimum triangulations for some special classes of 3-dimensional convex polyhedra. (1) For polyhedra without 3-cycles and degree-4 vertices we achieve a tight approximation ratio 3/2. (2) For polyhedra with vertices of degree-5 or above, we achieve an upper bound 2 - 1/12 on the approximation ratio. (3) For polyhedra with n vertices and vertex degrees bounded by a constant Δ we achieve an asymptotic tight ratio 2 - Ω(1/Δ) - Ω(1/n).