We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, 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 Seb\H{o} (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 $G$.

翻译：我们考虑以下称为最短两条不相交路径问题：给定一个无向图$G=(V,E)$，边权重为$w:E \rightarrow \mathbb{R}$，以及$G$中的两个终端$s$和$t$，找出$s$和$t$之间两条内部顶点不相交的路径，使得总权重最小。如Schlotter和Seb\H{o}（2022）最近所示，如果边可以具有负权重，即使权重函数是保守的（即$G$中不存在负权重环），该问题也会变为NP困难。我们提出一个多项式时间算法，用于在负权重边构成$G$中单棵树的情况下求解保守权重下的最短两条不相交路径问题。