the symmetric case) are achievable.

#### Our contributions.

We show algorithmic results for grid-like and tree-like input graphs as well as parameterized hardness for very restricted cases. summarizes our results. We discuss the different parameters subsequently.

[ l X l ]{} & result & reference\
& Dynamic programming solving <span style="font-variant:small-caps;">MinPAC</span> in &\
& $O(c^2 \cdot 2^c \cdot n + m + 4^c \cdot c^{2c - 3/2})$ time. &\
& An $O(\min(\{ x, \log_q n \}) \cdot n + m)$-time data reduction resulting in an equivalent <span style="font-variant:small-caps;">MinPAC</span> instance with at most $(q + 1) ^{2 x} + x$ vertices.&\
& An exponential-size kernel with respect to $x+q$. & \[cor:expKernelVCq\]\
& Linear-time data reduction resulting in an equivalent <span style="font-variant:small-caps;">MinPAC</span> instance with at most $20g-20$ vertices and $42g - 42$ arcs. &\
& A polynomial-size kernel with respect to $g$. & \[cor:poly-kernel-fes\]\
& <span style="font-variant:small-caps;">PAC</span> is NP-hard for any $h \geq 1$. &\
& <span style="font-variant:small-caps;">PAC</span> is W\[2\]-hard parameterized by $k$, even if the arcs have only cost zero or one and $h = 1$. &\
& <span style="font-variant:small-caps;">PAC</span> is not solvable in $2^{o(n)}$ time (assuming ETH).&\

It is known that the alignment of nodes in some regular grid-like patterns 