Convex Approximations for a Class of Integer Recourse Models: A Uniform Error Bound

We discuss the performance of the convex approximations introduced by Van der Vlerk [2004] for the class of integer recourse problems with totally unimodular (TU) recourse matrices. We show that the main result in Van der Vlerk [2004] needs stronger assumptions, so that a performance guarantee for the convex approximations is lacking in general. In order to obtain such a performance guarantee, we first analyze the approximations for simple integer recourse models. Using a new approach we improve the existing error bound for these models by a factor 2. We use insights from this analysis to obtain an error bound for complete integer recourse problems with TU recourse matrices. This error bound ensures that the performance of the approximations is good as long as the total variations of the densities of all random variables are small.