Using a Geometric Lens to Find \(\boldsymbol{k}\)-Disjoint Shortest Paths.

Given an undirected -vertex graph and pairs of terminal vertices, the -Disjoint Shortest Paths (-SDP) problem asks whether there are pairwise vertex-disjoint paths such that is a shortest --path for each . Recently, Lochet [Proceedings of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA ’21), SIAM, 2021, pp. 169–178] provided an algorithm that solves -SDP in time, answering a 20-year old question about the computational complexity of -SDP for constant . On the one hand, we present an improved -time algorithm based on a novel geometric view on this problem. For the special case on -edge graphs, we show that the running time can be further reduced to by small modifications of the algorithm and a refined analysis. On the other hand, we show that -SDP is W[1]-hard with respect to , showing that the dependency of the degree of the polynomial running time on the parameter is presumably unavoidable.