FPT factor approximation for cycle rank

Determine whether there exists an FPT-time factor-$g(k)$ approximation algorithm for the cycle rank of a digraph.

Background

Beyond exact FPT computation, the paper identifies as unresolved even the weaker possibility of an FPT-time approximation whose multiplicative factor depends only on the parameter. The authors explain that their constructive proof comes close to such an algorithm, but one Erdős–Pósa-type argument prevents them from establishing it.

References

Even the possibility of having an FPT-time factor-$g(k)$ approximation algorithm remains an interesting open problem.

— 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”