An improved FPT algorithm for the flip distance problem

Abstract

Given a set P of points in the Euclidean plane and two triangulations of P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O * (k •c k ) (c ≤ 2×14 11 ), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By applying the backtracking strategy and analyzing the underlying property of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O * (k • 32 k ).

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

“…It could also be interesting to study the possibility of reduction rules that reduce the number of non-common diagonals in an instance, since so far all work undertaken has been done by reducing the common diagonals. We also note that there has been quite sophisticated parameterized complexity work undertaken on the flip distance problem in recent years, although most of it has been done on more general versions of it: in triangulations of point sets on the plane [5,8]. We wonder whether those results can be strengthened in our more restricted setting i.e.…”

“…However, when translated to flip distance, the worst case corresponds to d subinstances, each of which has exactly one noncommon diagonal (and no common diagonals). Such subinstances are squares, and in the vast majority of the literature the size of the polygons is regarded as the number of outer edges [5,7,8,12]. Taking that metric, Lucas' kernel would yield 4d ≤ 4k for flip distance, not 2k, so the kernel distorts when using the usual sizes of the problems.…”

“…We note that by using the recent FPT algorithm of [5] instead of the trivial exponential-time algorithm, we can solve an instance of size less than 1/ in time O(32 1/ • poly(1/ )) instead of O((1/ ) 2/ ), which is a significant improvement.…”