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.
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