Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

Let κ_G(s, t) denote the maximum number of pairwise internally disjoint st-paths in a graph G = (V, E). For a set T⊆ V of terminals, G is k-T-connected if κ_G(s, t) ≥ k for all s, t ∈ T; if T = V then G is k-connected. Given a root node s, G is k- (T, s)-connected if κ_G(t, s) ≥ k for all t ∈ T. We consider the corresponding min-cost connectivity augmentation problems, where we are given a graph G = (V, E) of connectivity k, and an additional edge set Ê on V with costs. The goal is to compute a minimum cost edge set J⊆ Ê such that G∪ J has connectivity k + 1. For the k-T-Connectivity Augmentation problem when Ê is an edge set on T we obtain ratio O(ln|T||T|−k), improving the ratio O(|T||T|−k⋅ln|T||T|−k) of Nutov (Combinatorica, 34(1), 95–114, 2014). For the k-Connectivity Augmentation problem we obtain the following approximation ratios. For n ≥ 3k − 5, we obtain ratio 3 for directed graphs and 4 for undirected graphs, improving the previous ratio 5 of Nutov (Combinatorica, 34(1), 95–114, 2014). For directed graphs and k = 1, or k = 2 and n odd, we further improve to 2.5 the previous ratios 3 and 4, respectively. For the undirected 2-(T, s)-Connectivity Augmentation problem we achieve ratio 423, improving the previous best ratio 12 of Nutov (ACM Trans. Algorithms, 9(1), 1, 2014). For the special case when all the edges in Ê are incident to s, we give a polynomial time algorithm, improving the ratio 41730 of Kortsarz and Nutov, (2015) and Nutov (Algorithmica, 63(1-2), 398–410, 2012) for this variant.