An O(3. 82 k) Time FPT Algorithm for Convex Flip Distance

Abstract

Let P be a convex polygon in the plane, and let T be a triangulation of P . An edge e in T is called a diagonal if it is shared by two triangles in 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 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 Convex Flip Distance problem asks if the flip distance between two given triangulations of P is at most k, for some given parameter k∈ N . It has been an important open problem to decide whether Convex Flip Distance is NP-hard. In this paper, we present an FPT algorithm for the Convex Flip Distance problem that runs in time O(3. 82 k) and uses polynomial space, where k is the number of flips. This algorithm significantly improves the previous best FPT algorithms for the problem.