AI 中文总结
针对三终端可达性保持最小边割问题，提出利用割主导分解树概率分布的多项式时间 \(O(\log n)\) 近似算法，通过扩展根树簇成辅助图并利用其结构特性及分解树割失真证明近似保证。
AI 中文摘要
在三终端可达性保持最小边割问题中，输入为带权无向图及终端 \(s_1\)、\(s_2\)、\(t\)。目标是删除最小代价边集，使 \(t\) 与 \(s_1\)、\(s_2\) 分离，同时保持 \(s_1\) 与 \(s_2\) 连通。本文给出多项式时间 \(O(\log n)\) 近似算法。该算法利用割主导分解树的概率分布，通过将根树簇扩展为其在原图中诱导的连通分量来克服障碍，这些分量构成节点加权辅助图，其最小节点加权路径产生连通可行源侧。基于根树簇所有连通分量的总图边界成本不大于相应树边容量这一结构观察，结合分解树的期望 \(O(\log n)\) 割失真证明了近似保证。
英文摘要
In the three-terminal Reachability-Preserving Minimum Edge Cut problem, the input is an undirected edge-weighted graph with terminals \(s_1,s_2,t\). The objective is to delete a minimum-cost set of edges that separates \(t\) from both \(s_1\) and \(s_2\), while preserving connectivity between \(s_1\) and \(s_2\). We give a polynomial-time \(O(\log n)\)-approximation algorithm. The algorithm uses a probabilistic distribution of cut-dominating decomposition trees. A direct transfer of a connected tree solution to the original graph is not valid because a connected tree cluster may induce a disconnected vertex set in the graph. We overcome this obstruction by expanding every rooted tree cluster into the connected components it induces in the original graph. These components form a node-weighted auxiliary graph. A minimum node-weighted path in this auxiliary graph produces a connected feasible source side. The main structural observation is that the total graph-boundary cost of all connected components of a rooted tree cluster is no greater than the capacity of the corresponding tree edge. This permits the auxiliary path to be compared with a tree cut separating an optimal preserved \(s_1\)-\(s_2\) path from \(t\). Combining this comparison with the expected \(O(\log n)\) cut distortion of the decomposition trees proves the approximation guarantee.
