Network Design with Weighted Degree Constraints

In an undirected graph G = (V,E) with a weight function w : E×V → Q+, the weighted degree dw(v;E) of a vertex v is defined as ∑ {w(e, v) | e ∈ E incident with v}. In this paper, we consider a network design problem with upper-bound on weighted degree of each vertex. Inputs of the problem are an undirected graph G = (V,E) with E = E1 ∪̇ E2 ∪̇ E3, weights w1 : E1 × V → Q+, μ : E2 → Q+, ν : E3 → Q+, an edge-cost c : E → Q, a skew supermodular set function f : 2 → N, and a degree-bound b : A → Q+. A solution consists of F ⊆ E, and weights wi : Fi × V → Q+ for i ∈ {2, 3}, where Fi stands for F ∩ Ei. It is defined to be feasible if the cut-size of U in (V, F ) is at least f(U) for U ⊂ V , w2(e, u) + w2(e, v) = μ(e) for e = uv ∈ F2, {w3(e, u), w3(e, v)} = {0, ν(e)} for e = uv ∈ F3, and dw1(v;F1) + dw2(v;F2) + dw3(v;F3) ≤ b(v) for each v ∈ V . The goal of this problem is to find a feasible solution that minimizes its cost ∑ e∈F c(e). Relaxing the constraints on weighted degree, we propose a bi-criteria approximation algorithm based on the iterative rounding, which has been successfully applied to the degreebounded spanning tree problem. Our algorithm computes a (2, 9+5θ)-approximate solution, where θ = max{b(u)/b(v), b(v)/b(u) | uv ∈ E2} if E2 6= ∅ and θ = 0 if E2 = ∅, where (α, β)approximate solution has the cost at most α times the optimal and the weighted degree of v at most βb(v). We also give a (1, 5 + 3θ)-approximation algorithm to the case of f(U) = 1 for U ⊂ V . Moreover, a problem minimizing the maximum weighted degree of vertices is also discussed.