Flip Distance Between Two Triangulations of a Point-Set is NP-complete

TL;DR

This paper proves that calculating the minimum flip distance between two triangulations in complex planar point sets and polygons with holes is NP-complete, extending the known difficulty from convex polygons.

Contribution

It establishes NP-completeness for two natural generalizations of the flip distance problem, which was previously unresolved.

Findings

NP-completeness for polygons with holes

NP-completeness for point sets in the plane

Extends understanding of flip distance computational complexity

Abstract

Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.