Approximating Directed Weighted-Degree Constrained Networks

Given a graph H = (V, F ) with edge weights {we : e ∈ F}, the weighted degree of a node v in H is ∑ {wvu : 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) with edge-costs {ce : e ∈ E} and edge-weights {we : e ∈ E}, an intersecting supermodular set-function f on V , and degree bounds {b(v) : v ∈ B ⊆ 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 ⋅ opt, so that the weighted degree of every v ∈ V is at most: 7b(v) for arbitrary f and 5b(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, 4b(v))-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Similar results are shown for crossing supermodular f . We also consider the problem of packing maximum number k of pairwise 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⌋. Finally, for unit weights and without trying to bound the cost, we give an algorithm that computes a subgraph so that the degree of every v ∈ V is at most b(v) + 3, improving over the approximation b(v) + 4 of [2]. ∗Preliminary version in APPROX 2008, pp. 219–232.