Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G = (V,E), a family W of sets W ⊆ V of vertices (areas), and a requirement function r : W → Z+ (where Z+ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W ) edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area W ∈ W. So far this problem was shown to be NP-hard in the uniform case of r(W ) = 1 for each W ∈ W, and polynomially solvable in the uniform case of r(W ) = r ≥ 2 for each W ∈ W. In this paper, we show that the problem can be solved in O(m+ pn4 (r∗ + log n)) time, even if r(W ) ≥ 2 holds for each W ∈ W, where n = |V |, m = |{{u, v}|(u, v) ∈ E}|, p = |W|, and r∗ = max{r(W ) | W ∈ W}.