Flip Distance Is in FPT Time O(n+ k * c^k)

Abstract

Let T be a triangulation of a set P of n points in the plane, and let e be an edge shared by two triangles in T such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of P from T. The flip distance between two triangulations of 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 P is k, for some given k \in \mathbb{N}. It is a fundamental and a challenging problem.

In this paper we present an algorithm for the 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. 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 our algorithm.