An $O(\log n)$-Approximation for Three-Terminal Reachability-Preserving Minimum Edge Cut

Qi Duan

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arXiv · 2607.20114

An $O(\log n)$-Approximation for Three-Terminal Reachability-Preserving Minimum Edge Cut

Qi Duan

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Abstract

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.

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Qi Duan. 2026-07-22. An $O(\log n)$-Approximation for Three-Terminal Reachability-Preserving Minimum Edge Cut. https://arxiv.org/abs/2607.20114

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