Augmenting local edge-connectivity between vertices and vertex subsets in undirected graphs

Abstract

Given an undirected multigraph G = (V, E), a family W of sets W ⊆ V of vertices (areas), and a requirement function rW : W → Z+ (where Z+ is the set of positive integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least rW(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 rW(W) = 1 for each W ∈ W, and polynomially solvable in the uniform case of rW(W) = r ≥ 2 for each W ∈ W. In this paper, we show that the problem can be solved in O(m+ pr*n5 log(n/r*)) time, even in the general case of rW(W) ≥ 3 for each W ∈ W, where n = |V|, m = |{{u, v}|(u, v) ∈ E}|, p = |W|, and r* = max{rW(W) | W ∈ W}. Moreover, we give an approximation algorithm which finds a solution with at most one surplus edges over the optimal value in the same time complexity in the general case of rW(W) ≥ 2 for each W ∈ W. © Springer-Verlag Berlin Heidelberg 2003.