Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

TL;DR

This paper proves that finding the minimum number of edge flips to transform one triangulation of a planar point set into another is APX-hard, indicating the problem's computational difficulty.

Contribution

It establishes the APX-hardness of the flip distance problem between triangulations of planar point sets, a fundamental question in computational geometry.

Findings

Flip distance problem is APX-hard.

Transforming one triangulation to another via edge flips is computationally difficult.

Provides complexity bounds for triangulation transformations.

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 $T_2$.