Augmenting Edge-Connectivity between Vertex Subsets

Abstract

Given a graph G = (V, E) and a requirement function r: W1 x W2 → R+ for two families W1, W2 ⊆ 2V - {θ}, 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 G satisfies δG (X) ≥ r(W1, W2) for all X ⊆ V, W1 ∈ W1, and W2 ∈ W2 with W1 ⊆ X ⊆ V - W2, where δG(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 clog α(W1, W2), unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation, where α(W1, W2) denotes the number of pairs W1 ∈ W1 and W2 ∈ W2 with r(W1, W2) > 0. We also provide an O(log α (W1, W2))-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 (W1, W2))-approximable, unless P=NP, which solves open problems for node-to-area edge-connectivity augmentation (Ishii et al. 2008, Ishii and Hagiwara 2006, Miwa and Ito 2004). Furthermore, we characterize the node-to-area and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and extended modulotone functions.