Minimum Average Distance Triangulations (1112.1828v3)
Abstract: We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x and y along edges of T. The length of a path is the sum of the weights of its edges. Edge weights are assumed to be given as part of the problem for every pair of distinct points (x,y) in S2. In a different variant of the problem, the points are vertices of a simple polygon and we look for a triangulation of the interior of the polygon that is optimal in the same sense. We prove that a general formulation of the problem in which the weights are arbitrary positive numbers is strongly NP-complete. For the case when all the weights are equal we give polynomial-time algorithms. In the end we mention several open problems.