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} (2023), 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：我们考虑如下被称为“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$G$中的两个端点$s$和$t$，要求找到$s$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}（2023）最近所示，若边可具有负权重，即使权重函数是保守的（即图$G$中不存在负权重的环），该问题也会变为NP难。我们提出一种多项式时间算法，在负权重边构成$G$中单一树的情况下，求解保守权重的“最短两条不相交路径”问题。