using some (multi-hop) route through the network. Formally, we study the following optimization problem.

<span style="font-variant:small-caps;"><span style="font-variant:small-caps;">Min-Power Asymmetric Connectivity (MinPAC)</span></span>

[@lX@]{} **Input:** & A strongly connected directed graph $G$ and a weight (cost) function $w\colon A(G) \to {\mathds{N}}$.\
**Task:** & Find a strongly connected spanning subgraph $H$ of $G$ minimizing $$\sum_{v \in V} \max_{vu \in A(H)} w(vu).$$

#### Related work.

This problem was initially formalized and shown to be NP-complete by Chen and Huang [@chen1989strongly]. Since then, there have been numerous publications on polynomial-time approximation algorithms (an asymptotically optimal $O(\log n)$ approximation [@Calinescu2003], a constant approximation factor with symmetric arc weights [@chen1989strongly; @Calinescu2013approximate], approximation algorithms for special cases [@Carmi2007; @chen2005dual; @Calinescu2016]), and hardness results for special cases [@clementi2004power; @Carmi2007]. To the best of our knowledge, the parameterized complexity of <span style="font-variant:small-caps;">MinPAC</span> has not been investigated yet.

In previous work, we investigated the parameterized complexity of the symmetric version of our problem [@BBNN17]; the difference to MinPAC is that an undirected graph is given and for every undirected edge in the solution subgraph $H$ both endpoints pay at least the weight of the edge. The asymmetric case turns out to be more involved on a technical level. However, comparable results (to 