A half-integral Erd\H{o}s-P\'osa theorem for directed odd cycles

We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every directed graph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a set of at most $f(k)$ vertices meeting all directed odd cycles. We also give a polynomial-time algorithm for fixed $k$ which outputs one of the two outcomes. Using this algorithmic result, we give a polynomial-time algorithm for fixed $k$ to decide whether such $k$ directed odd cycles exist, or there are no $k$ vertex-disjoint directed odd cycles. This extends the half-integral Erd\H{o}s-P\'osa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.