Approximating directed weighted-degree constrained networks

Given a graph H = (V,F) with edge weights {w(e):e ∈ F}, the weighted degree of a node v in H is ∑ {w(vu):vu ∈ F}. We give bicriteria approximation algorithms for problems that seek to find a minimum cost directed graph that satisfies both intersecting supermodular connectivity requirements and weighted degree constraints. The input to such problems is a directed graph G = (V,E), edge-costs {c(e):e ∈ E}, edge-weights {w(e):e ∈ E}, an intersecting supermodular set-function f on V, and degree bounds {b(v):v ∈ V}. The goal is to find a minimum cost f-connected subgraph H = (V,F) (namely, at least f(S) edges in F enter every S ⊆ V) of G with weighted degrees ≤ b(v). Our algorithm computes a solution of cost ≤ 2, so that the weighted degree of every v ∈ V is at most: 7 b(v) for arbitrary f and 5 b(v) for a 0,1-valued f; 2b(v) + 4 for arbitrary f and 2b(v) + 2 for a 0,1-valued f in the case of unit weights. Another algorithm computes a solution of cost ≤ 3.opt and weighted degrees ≤ 6b(v). We obtain similar results when there are both indegree and outdegree constraints, and better results when there are indegree constraints only: a (1,4)-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Finally, we consider the problem of packing maximum number k of edge-disjoint arborescences so that their union satisfies weighted degree constraints, and give an algorithm that computes a solution of value at least [k/36].