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 , 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 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 Bansal et al. (2008) [2].