A Constant Factor Approximation for Minimum λ-Edge-Connected k-Subgraph with Metric Costs

In the (k, λ)-subgraph problem, we are given an undirected graph G = (V, E) with edge costs and two positive integers k and λ and the goal is to find a minimum cost simple λ-edge-connected subgraph of G with at least k nodes. This generalizes several classical problems, such as the minimum cost k-Spanning Tree problem or k-MST (which is a (k, 1)-subgraph), and minimum cost λ-edge-connected spanning subgraph (which is a (|V (G)|, λ)-subgraph). The only previously known results on this problem [12, 4] show that the (k, 2)-subgraph problem has an O(log n)-approximation (even for 2-node-connectivity) and that the (k, λ)-subgraph problem in general is almost as hard as the densest k-subgraph problem [12]. In this paper we show that if the edge costs are metric (i.e. satisfy triangle inequality), like in the k-MST problem, then there is an O(1)-approximation algorithm for (k, λ)-subgraph problem. This essentially generalizes the k-MST constant factor approximability to higher connectivity.