A tight $$\sqrt{2}$$ 2 -approximation for linear 3-cut

We investigate the approximability of the linear 3-cut problem in directed graphs. The input here is a directed graph $$D=(V,E)$$ with node weights and three specified terminal nodes $$s,r,t\in V$$ , and the goal is to find a minimum weight subset of non-terminal nodes whose removal ensures that s cannot reach r and t, and r cannot reach t. The precise approximability of linear 3-cut has been wide open until now: the best known lower bound under the unique games conjecture (UGC) was 4 / 3, while the best known upper bound was 2 using a trivial algorithm. In this work we completely close this gap: we present a $$\sqrt{2}$$ -approximation algorithm and show that this factor is tight under UGC. Our contributions are twofold: (1) we analyze a natural two-step deterministic rounding scheme through the lens of a single-step randomized rounding scheme with non-trivial distributions, and (2) we construct integrality gap instances that meet the upper bound of $$\sqrt{2}$$ . Our gap instances can be viewed as a weighted graph sequence converging to a “graph limit structure”. We complement our results by showing connections between the linear 3-cut problem and other fundamental cut problems in directed graphs.