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

TL;DR

This paper proves that determining flip distances in edge-labeled triangulations is computationally hard, specifically APX-hard for point sets and NP-hard for simple polygons, even when ignoring labels.

Contribution

It extends hardness results to the edge-labeled setting, confirming the complexity of flip distance problems in this context.

Findings

Flip distance is APX-hard for edge-labeled triangulations of point sets.

Flip distance is NP-hard for edge-labeled triangulations of simple polygons.

Hardness persists even when source and target triangulations are identical ignoring labels.

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.