In the integrated network design and scheduling (INDS) problem, we are asked to repair edges in a graph by using parallel machines so that the performance of the network is recovered by a certain level, and the objective is to minimize the makespan required to finish repairing edges. The main aim of this note is to show that polynomial-time approximation schemes exist for various families of the INDS problems with a constant number of parallel machines, including the problems associated with minimum spanning tree, shortest path, maximum edge-disjoint paths, and maximum-weight matching.

This short note gives an elementary alternative proof for a theorem of Danilov and Koshevoy on Minkowski summation and unimodularity in discrete convex analysis. It is intended to disseminate this fundamental theorem and make its proof accessible to researchers in optimization and operations research.