Minimum weight triangulation is NP-hard

A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1IN-3-SAT. The correct working of the gadgets is established with computer assistance, using geometric inclusion and exclusion criteria for MWT edges, such as the diamond test and the LMT-Skeleton heuristic, as well as dynamic programming on polygonal faces. General Terms Algorithms, Theory Keywords Optimal triangulations, PLANAR-1-IN-3-SAT Categories and Subject Descriptors F.2.2 [Nonnumerical Algorithms and Problems]: Geometrical problems and computations; G.2.2 [Graph Theory]: Graph algorithms

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Wolfgang Mulzer, Günter Rote
• Keywords: COMPGEOM 2006, Keywords Optimal Triangulations, Minimum Weight Triangulation, Plane Straight-line Graph