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}.

Augmenting Edge-Connectivity between Vertex Subsets

“…The function p defined in LAP in undirected graphs and NAAP with r depending only on W 2 in undirected graphs are not symmetric supermodular, but symmetric skew-supermodular, as observed by Frank (1992), Ishii and Hagiwara (2006). Since the assumption holds for these cases, the result of Nutov (2005) implies the 7/4-approximability for NAAP with r depending only on W 2 in undirected graphs.…”

“…This shows the inapproximability of NAAP, whose complexity status was left open , Ishii and Hagiwara 2006, Miwa and Ito 2004). Here we remark that the function p discussed by Benczúr and Frank (1999), Nutov (2005) is not skewsupermodular for these cases, and hence Nutov's result is not applicable.…”

“…We also provide an O(log α(W 1 , W 2 ))-approximation algorithm for the area-to-area edge-connectivity augmentation problem. This together with the negative result implies that the problem is Θ(log α(W 1 , W 2 ))-approximable, unless P=NP, which solves open problems for nodeto-area edge-connectivity augmentation , Ishii and Hagiwara 2006, Miwa and Ito 2004.…”

“…Furthermore, Ishii and Hagiwara (2006) showed that it is solvable in polynomial time, if a requirement function r depends only on W 2 and r ≥ 2. From the results of Ishii and Hagiwara (2006) and Nutov (2005), we can observe that NAAP for undirected graphs with r depending only on W 2 is 7/4-approximable in polynomial time. However, no other approximation result is known for NAAP in undirected/directed graphs.…”

“…As for LAP, Frank (1992Frank ( , 1994 showed that it ). * * 7/4-approximable if r depends only on W 2 (Nutov 2005), while solvable in O(n 3 |W 2 |(m + n log n)) time if in addition r ≥ 2 holds (Ishii and Hagiwara 2006). …”