Fixed-parameter tractability of cycle-rank computation

Determine whether computing the cycle rank of a digraph admits a fixed-parameter tractable algorithm parameterized by the cycle rank.

Background

Prior work gives an XP algorithm for deciding whether a digraph has cycle rank at most a fixed parameter and a polynomial-time approximation algorithm. The authors note that the existence of an FPT algorithm remains unresolved; their proof is largely constructive but contains an Erdős–Pósa-type step whose algorithmic complexity is not settled.