Shortest Two Disjoint Paths in Conservative Graphs

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph G = ( V, E) with edge weights w : E -> R, two terminals s and t in G, find two internally vertex-disjoint paths between s and t with minimum total weight. As shown recently by Schlotter and Sebo (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, i.e., there are no cycles in G with negative total weight. We propose a polynomial-time algorithm that solves the SHORTEST TWO DISJOINT PATHS problem for conservative weights in the case when the negative-weight edges form a constant number of trees in G.