Progress on Maximum Weight Triangulation

Abstract

In this paper, we investigate the maximum weight triangulation of a point set in the plane. We prove that the weight of maximum weight triangulation of any planar point set with diameter D is bounded above by $((2\epsilon+2) \cdot n + \frac{\pi(1-2\epsilon)}{8\epsilon\sqrt{1-\epsilon^2}} + \frac{\pi}{2} -- 5(\epsilon +1)) D$, where ε for $0 < \epsilon \leq \frac{1}{2}$ is a constant and n is the numb er of points in the set. If we use ‘spoke-scan’ algorithm to find a triangulation of the point set, we obtain an approximation ratio of 4.238. Furthermore, if the point set forms a ‘semi-lune’ or a ‘semi-circled’ convex polygon’, then its maximum weight triangulation can be found in O(n ²) time.