The minimum weight triangulation problem with few inner points

We look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n3) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(k6n5logn) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k=O(logn). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k inner points.