The Generalized Terminal Backup Problem

We consider the following network design problem, that we call the Generalized Terminal Backup Problem: given a graph (or a hypergraph) G0 = (V,E0), a set of (at least 2) terminals T ⊆ V and a requirement r(t) for every t ∈ T , nd a multigraph G = (V,E) such that λG0+G(t, T − t) ≥ r(t) for any t ∈ T . In theminimum cost version the objective is to nd G minimizing the total cost c(E) = ∑ uv∈E c(uv), given also costs c(uv) ≥ 0 for every pair u, v ∈ V . In the degree-speci ed version the question is to decide whether such a G exists, satisfying that the number of edges is a prescribed value m(v) at each node v ∈ V . The Terminal Backup Problem solved in [1] is the special case where G0 is the empty graph and r(t) = 1 for every terminal t ∈ T . We solve the Generalized Terminal Backup Problem in the following two cases. In the rst case we solve the degree-speci ed version by a splitting-o theorem. This splitting-o theorem in turn provides the solution for the minimum cost version in the case when c is node-induced, that is c(uv) = w(u) + w(v) for some node weights w : V → R+. In the second solved case we turn to the general minimum cost version, and we are able to solve it when G0 is the empty graph. This includes the Terminal Backup Problem [1] (r ≡ 1) and the Maximum-Weight b-matching Problem (T = V ). The solution depends on an interesting new variant of a theorem of Lovász and Cherkassky, and on the solution of the so-called Simplex Matching problem [1]. Our algorithms run in strongly polynomial time for both problems.