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, 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 $G$.

翻译：我们考虑以下称为“最短两条不相交路径”问题：给定一个带边权$w:E \rightarrow \mathbb{R}$的无向图$G=(V,E)$，以及$G$中的两个终端$s$和$t$，需找到$s$和$t$之间两条内部顶点不相交的路径，使得总权重最小。正如Schlotter和Seb\H{o}（2022）最近所证明，若边权可为负，即使权重函数是保守的（即$G$中不存在总权重为负的环），该问题将变为NP难问题。我们提出一种多项式时间算法，当负权边在$G$中构成常数棵树时，该算法可求解保守权重情形下的最短两条不相交路径问题。