Arboricity and subgraph listing algorithms
SIAM Journal on Computing
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 3-polytopes is NP-hard. The best approximation algorithms only give an approximation ratio of 2 for this problem, which is the best possible asymptotically when only combinatorial structures of the polytopes are considered. In this paper we improve the approximation ratio of finding minimum triangulations for some special classes of 3-dimensional convex polytopes. (1) For polytopes without 3-cycles and degree-4 vertices we achieve a tight approximation ratio of 3/2. (2) For polytopes where all vertices have degrees at least 5, we achieve an upper bound of 2 - 1/12 on the approximation ratio. (3) For polytopes with n vertices and vertex degrees bounded above by A we achieve an asymptotic tight ratio of 2 - Ω (1/Δ) - Ω(Δ/n). When Δ is constant the ratio can be shown to be at most 2 - 2/(Δ + 1).