Efficient Algorithms for Disjoint Shortest Paths Problem and its Extensions

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs $(s_1,t_1)$ and $(s_2,t_2)$, decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an $O(mn \log n)$ time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known $O(m^5n)$ time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in $O(mn^2 \log n)$ time. In addition, we extend our techniques to a more general setting: given two terminal pairs $(s_1, t_1)$ and $(s_2, t_2)$ in a directed graph, find the minimum possible number of vertex intersections between any shortest path from $s_1$ to $t_1$ and $s_2$ to $t_2$. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an $O(m^2 n^3)$ time algorithm for directed graphs with positive edge weights, and an $O(m+n)$ time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same $O(m+n)$ time. This is somewhat surprising, as there is no known $o(mn)$ time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).