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 subset of 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 is an element of V and an area W is an element of W. So far this problem was shown to be NP-hard in the uniform case of r(W) = 1 for each W is an element of W, and polynomially solvable in the uniform case of r(W) = r >= 2 for each W is an element of 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 is an element of W, where n = vertical bar V vertical bar, m = vertical bar{{u, v})vertical bar(u, v) is an element of E}vertical bar, p = vertical bar W vertical bar, and r* = max{r(W)vertical bar W is an element of W}. (c) 2006 Elsevier B.V. All rights reserved.