Mathematical Programming Manuscript No. on Splittable and Unsplittable Ow Capacitated Network Design Arc{set Polyhedra

We study the polyhedra of splittable and unsplittable single arc{set relaxations of multicommodity ow 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 19] and show that the separation problem of c{strong inequalities 7] is NP{hard, but can be solved over the subspace of fractional variables only. We introduce two classes of inequalities for the unsplittable ow problems 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 to test the eeectiveness of the results presented here empirically.