Max-Flow: Modelling edge dependencies

Question

For a toy problem I'm working on, I'd like to know if its possible to (easily) add the following type of constraints to a max-flow problem: if edge $x_{ij}$ is active (flow > 0), then also $x_{jk}$ must have flow. I.e. I'd like introduce an all-or-nothing type of constraint for subgraphs of the problem.

Answer 1

Suppose $u_{ij}$ is a constant upper bound on $x_{ij}$ and $\epsilon$ is a small positive constant. You can enforce $x_{ij}>0 \implies x_{jk} \ge \epsilon$ by introducing binary decision variable $y_{ij}$ and linear constraints \begin{align} x_{ij} &\le u_{ij} y_{ij} \\ x_{jk} &\ge \epsilon y_{ij} \end{align}