Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete

TL;DR

This paper proves that finding the shortest sequence of flips to transform one triangulation of a convex polygon into another, equivalent to rotation distance in binary trees, is NP-hard, resolving a long-standing open problem.

Contribution

It establishes the NP-hardness of computing minimum flip sequences in convex polygon triangulations and binary tree rotation distances, introducing new techniques for analyzing flip sequences.

Findings

Computing shortest flip sequences is NP-hard.

Rotation distance of binary trees is NP-hard.

Developed new techniques for flip sequence analysis.

Abstract

Flips in triangulations of convex polygons arise in many different settings. They are isomorphic to rotations in binary trees, define edges in the 1-skeleton of the Associahedron and cover relations in the Tamari Lattice. The complexity of determining the minimum number of flips that transform one triangulation of a convex point set into another remained a tantalizing open question for many decades. We settle this question by proving that computing shortest flip sequences between triangulations of convex polygons, and therefore also computing the rotation distance of binary trees, is NP-hard. For our proof we develop techniques for flip sequences of triangulations whose counterparts were introduced for the study of flip sequences of non-crossing spanning trees by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber~[SODA25] and Bjerkevik, Dorfer, Kleist, Ueckerdt, and Vogtenhuber~[SoCG26].