Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

TL;DR

This paper proves that calculating the minimum number of flips needed to transform one triangulation of a simple polygon into another is an NP-complete problem, highlighting its computational difficulty.

Contribution

It establishes the NP-completeness of the flip distance problem for simple polygons, extending the understanding of its computational complexity.

Findings

Flip distance computation is NP-complete for simple polygons.

The result complements previous APX-hardness findings for planar point sets.

The problem remains computationally hard even in the simple polygon case.

Abstract

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.