Approximating node connectivity problems via set covers

We generalize and unify techniques from several papers to obtain relatively simple and general technique for designing approximation algorithms for finding min-cost k-node connected spanning subgraphs. For the general instance of the problem, the previously best known algorithm has approximation ratio 2k. For k ≤ 5, algorithms with approximation ratio [(k+1)/2] are known. For metric costs Khuller and Raghavachari gave a (2 + 2(k−1)/n)-approximation algorithm. We obtain the following results. (i) An I(k−k₀)-approximation algorithm for the problem of making a k0-connected graph k-connected by adding a minimum cost edge set, where I(k) = 2+ (eqution found). (ii) A (2 + k−1/n)-approximation algorithm for metric costs. (iv) A [(k + 1)/2]-approximation algorithm for k = 6, 7. (v) A fast [(k + 1)/2]-approximation algorithm for k = 4. The multiroot problem generalizes the min-cost k-connected subgraph problem. In the multiroot problem, requirements k_u for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{k_u, k_v} internally disjoint paths between every pair of nodes u, v. For the general instance of the problem, the best known algorithm has approximation ratio 2k, where k = maxk_u. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for k ≤ 7 the approximation guarantee from 3 to 2 + [(k−1)/2]/k < 2.5.