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 ≥ 3 k − 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 42/3 , 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 417/30 of Kortsarz and Nutov, ( 2015 ) and Nutov (Algorithmica, 63 (1-2), 398–410, 2012 ) for this variant.