Computing the flip distance between triangulations

TL;DR

This paper introduces a fixed-parameter tractable algorithm for computing the minimum number of flips needed to transform one triangulation into another, advancing understanding of flip distance in planar graphs.

Contribution

The paper presents the first fixed-parameter tractable algorithm for the flip distance problem, with extensions to polygonal regions with holes and labeled triangulations.

Findings

Algorithm runs in time O(n + k * c^k) with c ≤ 2 * 14^{11}

Proves the flip distance problem is fixed-parameter tractable

Extends results to polygonal regions with holes and labeled graphs

Abstract

Let ${\cal T}$ be a triangulation of a set ${\cal P}$ of $n$ points in the plane, and let $e$ be an edge shared by two triangles in ${\cal T}$ such that the quadrilateral $Q$ formed by these two triangles is convex. A {\em flip} of $e$ is the operation of replacing $e$ by the other diagonal of $Q$ to obtain a new triangulation of ${\cal P}$ from ${\cal T}$. The {\em flip distance} between two triangulations of ${\cal P}$ is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of ${\cal P}$ is at most $k$, for some given $k \in N$. It is a fundamental and a challenging problem. We present an algorithm for the {\sc Flip Distance} problem that runs in time $O(n + k \cdot c^{k})$, for a constant $c \leq 2 \cdot 14^{11}$, which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.