Augmenting Local Edge-Connectivity between Vertices and Vertex Subsets in Undirected Graphs

Abstract

Given an undirected multigraph G=(V,E), a family ${\cal W}$ of sets W ⊆ V of vertices (areas), and a requirement function ${r_{{\cal W}}} : {\cal W}$ →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 ${r_{{\cal W}}}(W)$ edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area $W \in {\cal W}$. So far this problem was shown to be NP-hard in the uniform case of ${r_{{\cal W}}}(W)=1$ for each $W \in {\cal W}$, and polynomially solvable in the uniform case of ${r_{{\cal W}}}(W)=r \geq 2$ for each $W \in {\cal W}$. In this paper, we show that the problem can be solved in O(m + pr ^* n ⁵ log(n/r ^*)) time, even in the general case of ${r_{{\cal W}}}(W)\geq 3$ for each $W \in {\cal W}$, where n=|V|, m = |{{u,v}| (u,v) ∈ E}|, $p=|{\cal W}|$, and $r^*=\max\{{r_{{\cal W}}}(W)\mid W \in {\cal W}\}$. 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 ${r_{{\cal W}}}(W)\geq 2$ for each $W \in {\cal W}$.