On splittable and unsplittable flow capacitated network design arc-set polyhedra

We study the polyhedra of splittable and unsplittable single arc–set relaxations of multicommodity flow capacitated network design problems. We investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them. In particular, we give a linear–time separation algorithm for the residual capacity inequalities and show that the separation problem of c-strong inequalities is NP-hard, but can be solved over the subspace of fractional variables only. We introduce two classes of inequalities for the unsplittable flow problems. We present a summary of computational experiments with a branch-and-cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically.