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arXiv · 2610.34851

When is greedy exact for weighted minimum 3-cut?

Armando Rodado Amaris
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Abstract

The greedy algorithm for minimum 3-cut splits a global minimum-weight cut and then the cheaper of its two sides; it is a tight $4/3$-approximation. We show that it is exact, from every global minimum-weight cut, whenever some optimal 3-partition has two parts joined by total weight at least the weighted edge connectivity $λ$. This extends both theorems of Hochbaum and Shmoys from unweighted planar graphs to all weighted graphs, recovers the $4/3$ bound, and generalizes to minimum $k$-cut with a one-paragraph proof. The remaining hard instances have all pair weights below $λ$; for them we prove a merge lemma for Gomory-Hu sides. Finally, we give a 7-vertex graph with a unique Gomory-Hu tree whose unique optimal 3-partition contains no side of that tree, so the first stage of Kapoor's algorithm does not suffice on its own. The main theorem, the 4/3 bound, the merge lemmas and the counterexample are verified in the Lean 4 proof assistant.

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Armando Rodado Amaris. 2026-10-26. When is greedy exact for weighted minimum 3-cut?. https://arxiv.org/abs/2610.34851

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