  The Fr\'echet distance is a popular measure of dissimilarity for polygonal
curves. It is defined as a min-max formulation that considers all
direction-preserving continuous bijections of the two curves. Because of its
susceptibility to noise, Driemel and Har-Peled introduced the shortcut
Fr\'echet distance in 2012, where one is allowed to take shortcuts along one of
the curves, similar to the edit distance for sequences. We analyse the
parameterized version of this problem, where the number of shortcuts is bounded
by a parameter $k$. The corresponding decision problem can be stated as
follows: Given two polygonal curves $T$ and $B$ of at most $n$ vertices, a
parameter $k$ and a distance threshold $\delta$, is it possible to introduce
$k$ shortcuts along $B$ such that the Fr\'echet distance of the resulting curve
and the curve $T$ is at most $\delta$? We study this problem for polygonal
curves in the plane. We provide a complexity analysis for this problem with the
following results: (i) assuming the exponential-time-hypothesis (ETH), there
exists no algorithm with running time bounded by $n^{o(k)}$; (ii) there exists
a decision algorithm with running time in $O(kn^{2k+2}\log n)$. In contrast, we
also show that efficient approximate decider algorithms are possible, even when
$k$ is large. We present a $(3+\varepsilon)$-approximate decider algorithm with
running time in $O(k n^2 \log^2 n)$ for fixed $\varepsilon$. In addition, we
can show that, if $k$ is a constant and the two curves are $c$-packed for some
constant $c$, then the approximate decider algorithm runs in near-linear time.
