Summary: On Minimum Weight Pseudo-Triangulations
Oswin Aichholzer Franz Aurenhammer Thomas Hackl Bettina Speckmann§
In this note we discuss some structural properties of minimum weight pseudo-triangulations.
Optimal triangulations for a set of points in the plane have been, and still are, extensively studied
within Computational Geometry. There are many possible optimality criteria, often based on
edge weights or angles. One of the most prominent criteria is the weight of a triangulation, that
is, the total Euclidean edge length. Computing a minimum weight triangulation (MWT) for a
point set has been a challenging open problem for many years  and various approximation
algorithms were proposed over time; see e.g.  for a short survey. Mulzer and Rote  showed
only very recently that the MWT problem is NP-hard.
Pseudo-triangulations are related to triangulations and use pseudo-triangles in addition to
triangles. A pseudo-triangle is a simple polygon with exactly three interior angles smaller
than . Also for pseudo-triangulations several optimality criteria have been studied, for example,
concerning the maximum face or vertex degree . Optimal pseudo-triangulations can also be
found via certain polytope representations  or via a realization as locally convex surfaces in
three-space . Not all of these optimality criteria have natural counterparts for triangulations.
Here we consider the classic minimum weight criterion for pseudo-triangulations.
Rote et al.  were the first to ask for an algorithm to compute a minimum weight pseudo-