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++⌊k/2⌋-1 j=1 1/j ⌊k/j+1⌋. (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 ku for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{ku, kv} 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 = maxku. 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.