# On the complexity of finding internally vertex-disjoint long directed paths

## TL;DR

This paper investigates the computational complexity of detecting subdivisions of spindles in directed graphs, revealing polynomial solvability for small path lengths and NP-hardness otherwise, with specialized algorithms for certain cases.

## Contribution

It establishes a complexity dichotomy for finding spindle subdivisions, introduces FPT algorithms for two-path spindles, and analyzes the problem in acyclic graphs.

## Key findings

- Polynomial-time for $oldsymbol{ ext{l} ext{ } ext{≤} ext{ } 3}$
- NP-hard for larger $oldsymbol{ ext{l}}$
- FPT algorithms for two-path spindles

## Abstract

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.