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+ pn 4 (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}.
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