An Improved Kernel for the Flip Distance Problem on Simple Convex Polygons

The complexity of computing the flip distance between two triangulations of a simple convex polygon is unknown. Here we approach the problem from a parameterized complexity perspective and improve upon the 2k kernel of Lucas [10]. Specifically, we describe a kernel of size 43k and then show how it can be improved to (1+?)k for every constant ? > 0. By ensuring that the kernel consists of a single instance our result yields a kernel of the same magnitude (up to additive terms) for the almost equivalent rotation distance problem on rooted, ordered binary trees. The earlier work of Lucas left the kernel as a disjoint set of instances, potentially allowing very minor differences in the definition of the size of instances to accumulate, causing a constant-factor distortion in the kernel size when switching between flip distance and rotation distance formulations. Our approach avoids this sensitivity.