  For two positive integers $k$ and $\ell$, a $(k \times \ell)$-spindle is the
union of $k$ pairwise internally vertex-disjoint directed paths with $\ell$
arcs between two vertices $u$ and $v$. We are interested in the (parameterized)
complexity of several problems consisting in deciding whether a given digraph
contains a subdivision of a spindle, which generalize both the Maximum Flow and
Longest Path problems. We obtain the following complexity dichotomy: for a
fixed $\ell \geq 1$, finding the largest $k$ such that an input digraph $G$
contains a subdivision of a $(k \times \ell)$-spindle is polynomial-time
solvable if $\ell \leq 3$, and NP-hard otherwise. We place special emphasis on
finding spindles with exactly two paths and present FPT algorithms that are
asymptotically optimal under the ETH. These algorithms are based on the
technique of representative families in matroids, and use also color-coding as
a subroutine. Finally, we study the case where the input graph is acyclic, and
present several algorithmic and hardness results.
