Approximation of Minimum Triangulation for Polyhedron with Bounded Degrees

Abstract

Finding minimum triangulations of convex polyhedra is NPhard. 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).

This work is supported by RGC grant HKU 7019/00E.