Static and dynamic routing under the single-source Hose model

 This paper addresses special cases of the robust network design problem under the single-source Hose model. We show that, in the case of unitary bounds, the static and the dynamic routing approaches lead to the same optimal solution, and this is true for both the splittable and the unspittable scenarios. As a consequence, in such a special case, the robust network design problem with (splittable) dynamic routing is polynomially solvable, whereas the problem is coNP-Hard under the general single-source Hose model. The results are based on the fact that the single-source Hose polyhedron with unitary bounds is dominated by a polynomial number of demand vectors. A feasible static routing can then be constructed as a convex combination of a set of routing templates which are feasible for the dominant demand vectors. The equivalence between static and dynamic routing is a consequence of those results, and it can also be generalized to some single-source Hose cases with non unitary bounds.