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

2026-07-22

AI Summary

This work addresses the minimum edge cut problem for preserving three-terminal reachability in undirected weighted graphs: given terminals \(s_1, s_2, t\), the goal is to remove a minimum-cost set of edges such that \(t\) becomes disconnected from both \(s_1\) and \(s_2\), while \(s_1\) and \(s_2\) remain connected. The paper presents the first polynomial-time algorithm with a logarithmic approximation guarantee for this NP-hard problem. The key innovation lies in constructing a probability distribution over cut-dominating decomposition trees, expanding their tree clusters into connected components of the original graph to overcome mapping barriers, and showing that the capacity of tree edges can be upper-bounded by the total boundary cost of corresponding components. By building an auxiliary vertex-weighted graph and computing a minimum-weight path, the method yields a feasible solution with an \(O(\log n)\) approximation ratio.

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.

Problem

• Reachability-Preserving
• Minimum Edge Cut
• Three-Terminal
• Graph Connectivity
• Approximation Algorithm

Innovation

• approximation algorithm
• reachability-preserving cut
• decomposition tree
• auxiliary graph
• edge connectivity