vertex $v \in V(G)$, we write $N^+_G(v) = \{ u \mid vu \in A \}$ and $N^-_G(v) = \{ u \mid uv \in A \}$ to denote the out- and in-neighborhood of $v$. We define the degree of $v$ as $\deg_G(v) = |N^+_G(v) \mathbin{\cup} N^-_G(v)|$. We say that $S \subseteq V(G)$ is a strongly connected component if there exists a path from each vertex $u \in S$ to every other vertex $v \in S$ in $G[S]$. We write $\mathcal{S}_G$ to denote the set of strongly connected components. We use ${\ensuremath{U_{G}}}$ to denote the underlying undirected graph of $G$. We denote the optimal cost of an instance of <span style="font-variant:small-caps;">MinPAC</span> $I$ by $\operatorname{OPT}(I)$. The cost of a vertex subset $V' \subseteq V(G)$ in a solution with arcs $A' \subseteq A(G)$ is denoted by $\operatorname{Cost}(V', A', w) = \sum_{v \in V'} \max_{vu \in A'} w(vu)$. For ease of notation, we write $w(vu) = \infty$ if $vu \not\in A$.

A parameterized problem $\Pi$ is a set of pairs $(I, k)$, where $I$ denotes the problem instance and $k$ is the parameter. The problem $\Pi$ is *fixed-parameter tractable* (FPT) if there exists an algorithm solving any instance of $\Pi$ in $f(k) \cdot |I|^c$ time, 