Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes.