Stochastic Bounds for the Max Flow in a Network with Discrete Random Capacities

We show how to obtain stochastic bounds for the strong stochastic ordering and the concave ordering of the maximal flow in a network where the capacities are non negative discrete random variables. While the deterministic problem is polynomial, the stochastic version with discrete random variables is NP-hard. The monotonicity of the Min-Cut problem for these stochastic orderings allows us to simplify the input distributions and obtain bounds on the results. Thus we obtain a tradeoff between the complexity of the computations and the precision of the bounds. We illustrate the approach with some examples.