Flip Distance is in FPT time $O(n+ k \cdot c^k)$.
Iyad A. Kanj, Ge Xia · arXiv (Cornell University) · 2014
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 $k$, for some given $k \in N$. It is a fundamental and a challenging problem whose complexity for the case of triangulations of a convex polygon remains open for over 25 years. In this paper we present an algorithm for the Flip Distance problem that runs in time $O(n + k \cdot c^{k})$, for $c=392$, which implies that the problem is fixed-parameter tractable. The NP-hardness reduction for the Flip Distance problem given by Lubiw and Pathak can be used to show that, unless the exponential-time hypothesis (ETH) fails, the Flip Distance problem cannot be solved in time $O^*(2^{o(k)})$. Therefore, one cannot expect an asymptotic improvement in the exponent of the running time of the presented algorithm.