Approximating the minimum triangulation of convex 3-polytopes with bounded degrees

Abstract

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 Δ 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 ) .