An almost four-approximation algorithm for maximum weight triangulation

We consider the following planar maximum weight triangulation (MAT) problem: given a set of n points in the plane, find a triangulation such that the total length of edges in triangulation is maximized. We prove an Omega(root n) lower bound on the approximation factor for several heuristics: maximum greedy triangulation, maximum greedy spanning tree triangulation and maximum spanning tree triangulation. We then propose the Spoke Triangulation algorithm, which approximates the maximum weight triangulation for points in general position within a factor of almost four in O(n log n) time. The proof is simpler than the previous work. We also prove that Spoke Triangulation approximates the maximum weight triangulation of a convex polygon within a factor of two.