Shortest two disjoint paths in conservative graphs
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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 Sebő (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, i.e., there are are no cycles in G with negative 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 single tree in G.
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