# Network Design with Probabilistic Capacities

## TL;DR

This paper addresses a network design problem with probabilistic arc capacities, introducing novel formulations and algorithms that leverage supermodularity and submodularity to efficiently handle complex probabilistic constraints.

## Contribution

It presents a new formulation for probabilistic network design, along with separation procedures and cutting plane methods exploiting supermodularity and submodularity for both independent and correlated capacities.

## Key findings

- Exploiting supermodularity improves solution efficiency.
- Reformulation for correlated capacities enhances computational performance.
- Significant advantages over classical approaches are demonstrated.

## Abstract

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.