A Note on the Flip Distance Problem for Edge-Labeled Triangulations

Abstract: For both triangulations of point sets and simple polygons, it is known that determining the flip distance between two triangulations is an NP-hard problem. To gain more insight into flips of triangulations and to characterize "where edges go" when flipping from one triangulation to another, flips in edge-labeled triangulations have lately attracted considerable interest. In a recent breakthrough, Lubiw, Mas\'arov\'a, and Wagner (in Proc. 33rd Symp. of Computational Geometry, 2017) prove the so-called "Orbit Conjecture" for edge-labeled triangulations and ask for the complexity of the flip distance problem in the edge-labeled setting. By revisiting and modifying the hardness proofs for the unlabeled setting, we show in this note that the flip distance problem is APX-hard for edge-labeled triangulations of point sets and NP-hard for triangulations of simple polygons. The main technical challenge is to show that this remains true even if the source and target triangulation are the same when disregarding the labeling.