USING A GEOMETRIC LENS TO FIND K -DISJOINT SHORTEST PATHS

Given an undirected n-vertex graph and k pairs (s(1), t(1)),..., (s(k), t(k)) of terminal vertices, the k-Disjoint Shortest Paths (k-SDP) problem asks whether there are k pairwise vertexdisjoint paths P-1,..., P-k such that Pi is a shortest si-ti-path for each i is an element of [k]. Recently, Lochet [Proceedings of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA '21), SIAM, 2021, pp. 169-178] provided an algorithm that solves k-SDP in n(O(k5k)) time, answering a 20-year old question about the computational complexity of k-SDP 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-SDP 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.