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

• Toshimasa Ishii
• Masayuki Hagiwara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)

## 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 $$W \in {\cal W}$$.

## Keywords

Admissible Pair Requirement Function Uniform Case Area Graph Vertex Subset
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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