Augmenting Edge-Connectivity between Vertex Subsets

Abstract

Given a directed or undirected graph G=(V,E), a collection ${\mathcal{R}}=\{(S_{i},T_{i}) \mid i=1,2,\ldots,|{\mathcal{R}}|, S_{i},T_{i} \subseteq V, S_{i} \cap T_{i} =\emptyset\}$ of two disjoint subsets of V, and a requirement function $r: {\mathcal{R}} \to\mathbb{R}_{+}$, we consider the problem (called area-to-area edge-connectivity augmentation problem) of augmenting G by a smallest number of new edges so that the resulting graph $\hat{G}$ satisfies $d_{\hat{G}}(X)\geq r(S,T)$ for all X⊆V, $(S,T) \in{\mathcal{R}}$ with S⊆X⊆V−T, where d _G (X) denotes the degree of a vertex set X in G. This problem can be regarded as a natural generalization of the global, local, and node-to-area edge-connectivity augmentation problems.

In this paper, we show that there exists a constant c such that the problem is inapproximable within a ratio of $c\log{|{\mathcal{R}}|}$, unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation. We also provide an ${\mathrm{O}}(\log{|{\mathcal{R}}|})$-approximation algorithm for the area-to-area edge-connectivity augmentation problem, which is a natural extension of Kortsarz and Nutov’s algorithm (Kortsarz and Nutov, J. Comput. Syst. Sci., 74:662–670, 2008). This together with the negative result implies that the problem is ${\varTheta}(\log{|{\mathcal{R}}|})$-approximable, unless P=NP, which solves open problems for node-to-area edge-connectivity augmentation in Ishii et al. (Algorithmica, 56:413–436, 2010), Ishii and Hagiwara (Discrete Appl. Math., 154:2307–2329, 2006), Miwa and Ito (J. Oper. Res. Soc. Jpn., 47:224–243, 2004).

Furthermore, we characterize the node-to-area and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and k-modulotone functions.