A relaxation of the Directed Disjoint Paths problem: A global congestion metric helps

In the Directed Disjoint Paths problem, we are given a digraph $D$ and set of requests $\{(s_1, t_1), \ldots, (s_k, t_k)\}$, task is to find collection pairwise vertex-disjoint paths $\{P_1, P_k\}$ such that each $P_i$ path from $s_i$ $t_i$ in $D$. This problem NP-complete for fixed $k=2$ W[1]-hard with parameter $k$ DAGs. A few positive results known under restrictions on input digraph, as being planar or having bounded directed tree-width, relaxations allowing vertex congestion. Positive scarce, however, general digraphs. this article propose novel global congestion metric problem: only require be "disjoint enough", sense they must behave properly not whole graph, but an unspecified part size prescribed by parameter. Namely, Enough $n$-vertex $D$, requests, non-negative integers $d$ $s$, connecting at least vertices occur most $s$ collection. We study parameterized complexity number choices parameter, including tree-width Among other results, show DAGs and, side, give algorithm time $\mathcal{O}(n^{d+2} \cdot k^{d\cdot s})$ kernel $d 2^{k-s}\cdot \binom{k}{s} + 2k$ latter result has consequences Steiner Network it FPT terminals $p$, where $p = n - q$ $q$ solution.