where $f$ is some computable function and $c$ is some constant. A *reduction to a problem kernel* is a polynomial-time algorithm that, given an instance $(I, k)$ of $\Pi$, returns an equivalent instance $(I', k')$, such that $|I'| + k' \le g(k)$ for some computable function $g$. Problem kernels are usually achieved by applying *data reduction rules*. Given an instance $(I, k)$ for <span style="font-variant:small-caps;">MinPAC</span>, our data reduction rules compute in polynomial time a new instance $(I', k')$ of <span style="font-variant:small-caps;">MinPAC</span> and a number $d$. We call a data reduction rule *correct*, if $\operatorname{OPT}(I) = \operatorname{OPT}(I') + d$.

Parameterization by the number of strongly connected components induced by the obligatory arcs {#sec:fpt}
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In this section we present a fixed-parameter algorithm with respect to the number $c$ of strongly connected components (SCCs) induced by *obligatory arcs*—arcs that can be included into any optimal solution with no additional cost. We find the obligatory arcs by means of lower bounds on costs paid by each vertex.

A *vertex lower bound* is a function $\ell \colon V(G) \to {\mathds{N}}$ such that for any optimal solution $H$ and any vertex $v \in V(G)$, it holds that $$\ell(v) \le \max_{vu \in A(H)} w(vu).$$

Observe that 