On Splittable and Unsplittable Ow Capacitated Network Design Arc{set Polyhedra

We study the polyhedra of splittable and unsplittable arc sets of multicommodity ow capac-itated network design problems. We investigate the optimization problems over these polyhedra 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 (Magnanti et al., 1993) and show that the separation problem of c{strong inequalities (Brockmuller et al., 1996) is NP-hard, but can be solved over the subspace of fractional variables only. We introduce two new classes of inequalities that generalize the c{strong inequalities and show that the lifting of one of them can be done in polynomial time. We present a summary of computational experiments with a branch-and-cut algorithm for multicommodity ow capacitated network design problems illustrating the eeectiveness of the results presented here empirically.