Algorithmic implementation of the Erdős–Pósa step

Determine whether the Erdős–Pósa-type argument used in the cycle-rank proof can be converted into an FPT algorithm.

Background

The proof framework is described as mostly constructive and nearly sufficient for an FPT algorithm or FPT approximation algorithm for cycle rank. The unresolved issue is specifically whether the Erdős–Pósa-type argument used in the proof can be implemented within fixed-parameter running time.

References

Our proof is mostly constructive and comes close to yielding such an algorithm, but there is a part involving an Erd\H{o}s-Posa-type argument (\cref{thm:inductionmainstar}) for which it is unclear whether it can be turned into an FPT algorithm; see the sketch of the proof below for more details.

— Unavoidable butterfly minors in digraphs of large cycle rank  (2507.11814 - Hatzel et al., 16 Jul 2025) in Section 1, paragraph beginning “As mentioned before”