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Flip Distance is in FPT time O(n+ k * c^k)

Prof. Iyad Kanj (DePaul University)

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 N. It is a fundamental and a challenging problem whose complexity for the case of triangulations of a convex polygon remains open for over 25 years.

In this talk we present an algorithm for the Flip Distance problem that runs in time O(n + k * c^k), for a constant c <= 2 * 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.

Iyad Kanj
TEL 512

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