Computing the Minimum Weight Triangulation of a Set of Linearly Ordered Points

Consider a set of points, P, in the plane. A triangulation P is a partition of the plane obtained by joining the points in P with noncrossing straight line segments so that every region interior to the convex hull of P is a triangle. Although there are many polynomial time algorithms to obtain the triangulation of a set of points, there is no known polynomial time algorithm to obtain a triangulation that minimizes the sum of the edge lengths of the triangulation [Xl. Let us denote such a triangulation by MWT, that is, it is a minimum weight triangulation. This seemingly innocuous problem has proved to be one of the most perplexing problems in combinatorial optimization. Not only is there no polynomial time algorithm to obtain an MWT for an arbitrary set of points, the problem is also not known to be NP-hard [2]. Various aspects of this problem have been explored. A possible avenue of investigation is to look at approximation schemes. In this scenario a polynomial time approximation algorithm is pro-