  We consider a network design problem with random arc capacities and give a
formulation with a probabilistic capacity constraint on each cut of the
network. To handle the exponentially-many probabilistic constraints a
separation procedure that solves a nonlinear minimum cut problem is introduced.
For the case with independent arc capacities, we exploit the supermodularity of
the set function defining the constraints and generate cutting planes based on
the supermodular covering knapsack polytope. For the general correlated case,
we give a reformulation of the constraints that allows to uncover and utilize
the submodularity of a related function. The computational results indicate
that exploiting the underlying submodularity and supermodularity arising with
the probabilistic constraints provides significant advantages over the
classical approaches.
