Minimum augmentation of 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 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 } .