Using a geometric lens to find k disjoint shortest paths

Given an undirected $n$-vertex graph and $k$ pairs of terminal vertices $(s1,t1), \ldots, (sk,tk)$, the $k$-Disjoint Shortest Paths ($k$-DSP)-problem asks whether there are $k$ pairwise vertex-disjoint paths $P1,\ldots, Pk$ such that $Pi$ is a shortest $si$-$t_i$-path for each $i \in [k]$. Recently, Lochet [SODA 2021] provided an algorithm that solves $k$-DSP in $n^{O(k{5k})}$ time, answering a 20-year old question about the computational complexity of $k$-DSP for constant $k$. On the one hand, we present an improved $n^{{O(k!k)}$-time} algorithm based on a novel geometric view on this problem. For the special case $k=2$ on $m$-edge graphs, we show that the running time can be further reduced to $O(nm)$ by small modifications of the algorithm and a refined analysis. On the other hand, we show that $k$-DSP is W[1]-hard with respect to $k$, showing that the dependency of the degree of the polynomial running time on the parameter $k$ is presumably unavoidable.