Close the parameterized running-time gap for directed disjoint paths

Close the remaining gap between the upper bound for the $k$ directed vertex-disjoint paths problem with $k=omega(1)$ and the lower bound established for $k=o(tw^4)$, where $tw$ denotes treewidth.

Background

The paper studies the $k$ directed vertex-disjoint paths problem ($k$DDP) parameterized by the treewidth $tw$ of the input graph. It provides an algorithm with running time $2^{mathcal O((tw+k)\log k)}\cdot n$, which is faster than the previously known lower bound in the regime of sufficiently small $k$, including $k=tw^{o(1)}$.

The paper also establishes a conditional lower bound under the Strongly Exponential Time Hypothesis for general $k$: no $(2-\epsilon)^{pw\log pw}\cdot n^{\mathcal O(1)}$-time algorithm exists, where $pw$ is pathwidth. The stated open problem concerns the unresolved range between the upper-bound regime described as $k=\omega(1)$ and the lower-bound regime described as $k=o(tw^4)$.