Flip distance is in FPT time O (n + k [center dot]  c [superscript] k)

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.

Title	Flip distance is in FPT time O (n + k [center dot] c [superscript] k)
Creator	Kanj, I.
Creator	Xia, Ge
Publisher	32nd Symposium on Theoretical Aspects of Computer Science (STACS 2015)
Academic Department	Computer Science
Division	Natural Sciences
Organization	Lafayette College
Date Issued	2015
Date Available	2015-05-18T20:24:39Z
Type	Article
Language	English
Keyword	parameterized algorithms
	flip distance
	triangulations
Bibliographic Citation	Kanj, I. and G. Xia (2015) "Flip distance is in FPT time O (n + k [center dot] c [superscript] k)." In E. W. Mayr and N. Ollinger (Eds.) 2nd Symposium on Theoretical Aspects of Computer Science (STACS 2015) (pp 500–512). Germany: Schloss Dagstuhl--Leibniz-Zentrum fur Informatik
Standard Identifier	DOI 10.4230/LIPIcs.STACS.2015.500
Standard Identifier	Handle 10385/1923
Permalink	http://hdl.handle.net/10385/1923
Rights Statement	Creative Commons - Attribution
Rights Holders	Kanj, I.
Rights Holders	Xia, Ge