An $\mathcal {O}(3.82^{k})$ Time $\textsf {FPT}$ Algorithm for Convex Flip Distance

Abstract

Let $\mathcal{P}$ be a convex polygon in the plane, and let $\mathcal{T}$ be a triangulation of $\mathcal{P}$. An edge e in $\mathcal{T}$ is called a diagonal if it is shared by two triangles in $\mathcal{T}$. A flip of a diagonal e is the operation of removing e and adding the opposite diagonal of the resulting quadrilateral to obtain a new triangulation of $\mathcal{P}$ from $\mathcal{T}$. The flip distance between two triangulations of $\mathcal{P}$ is the minimum number of flips needed to transform one triangulation into the other. The Convex  Flip  Distance problem asks if the flip distance between two given triangulations of $\mathcal{P}$ is at most k, for some given parameter $k \in \mathbb {N}$. It has been an important open problem to decide whether Convex  Flip  Distance is NP-hard. In this paper, we present an $\textsf {FPT}$ algorithm for the Convex  Flip  Distance problem that runs in time $\mathcal {O}(3.82^{k})$ and uses polynomial space, where k is the number of flips. This algorithm significantly improves the previous best $\textsf {FPT}$ algorithms for the problem.