We study the undirected three-terminal reachability-preserving minimum edge cut problem. The input is an undirected graph $G=(V,E)$ with nonnegative edge costs, two protected terminals $s_1,s_2$, and a target terminal $t$. The goal is to remove a minimum-cost edge set so that $t$ is disconnected from the protected terminals while $s_1$ and $s_2$ remain connected. This problem captures a basic tension between separation and connectivity preservation. Prior work on connectivity-preserving cuts established polynomial-time solvability for some special cases, such as planar edge-cut instances, and strong hardness for node-cut variants, but a general-graph approximation guarantee for the undirected three-terminal edge-cut version does not appear to have been known. We give a polynomial-time $O(\sqrt n)$-approximation algorithm in this paper. This is the first known approximation algorithm for the problem

翻译：我们研究无向三终端可达性保持最小边割问题。输入为一个无向图$G=(V,E)$，其中边具有非负代价，两个受保护终端$s_1,s_2$，以及一个目标终端$t$。目标是移除一个最小代价边集，使得$t$与受保护终端断开连接，同时$s_1$与$s_2$保持连通。该问题刻画了分离性与连通性保持之间的基本张力。先前关于连通性保持割的研究确立了某些特殊情形（如平面边割实例）的多项式时间可解性，以及点割变体的强难解性，但无向三终端边割版本的一般图近似保证似乎尚未被知晓。本文给出一个多项式时间$O(\sqrt n)$近似算法。这是该问题已知的第一个近似算法。