Flip distance between triangulations of a planar point set is APX-hard

Abstract

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set S in the Euclidean plane and two triangulations $T_1$ and $T_2$ of S, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.