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].

翻译：凸多边形三角剖分中的翻转出现在许多不同场景中。它们与二叉树中的旋转同构，定义了Associahedron的1-骨架中的边，并构成了Tamari格中的覆盖关系。数十年来，确定将一个凸点集的三角剖分转换为另一个所需的最小翻转次数，其计算复杂度一直是一个令人困惑的开放性问题。我们通过证明计算凸多边形三角剖分之间的最短翻转序列（因此也计算二叉树的旋转距离）是NP困难的，从而解决了这个问题。在我们的证明中，我们发展了针对三角剖分翻转序列的技术，其对应方法由Bjerkevik、Kleist、Ueckerdt和Vogtenhuber~[SODA25]以及Bjerkevik、Dorfer、Kleist、Ueckerdt和Vogtenhuber~[SoCG26]在研究非交叉生成树的翻转序列时引入。