Preventing Small $\mathbf{(s,t)}$-Cuts by Protecting Edges

We introduce and study Weighted Min $(s,t)$-Cut Prevention, where we are given a graph $G=(V,E)$ with vertices $s$ and $t$ and an edge cost function and the aim is to choose an edge set $D$ of total cost at most $d$ such that $G$ has no $(s,t)$-edge cut of capacity at most $a$ that is disjoint from $D$. We show that Weighted Min $(s,t)$-Cut Prevention is NP-hard even on subcubcic graphs when all edges have capacity and cost one and provide a comprehensive study of the parameterized complexity of the problem. We show, for example W[1]-hardness with respect to $d$ and an FPT algorithm for $a$.