Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

Abstract

Let κ _G (s, t) denote the maximum number of pairwise internally disjoint st-paths in a graph G = (V, E). For a set $T \subseteq V$ 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 $\hat E$ on V with costs. The goal is to compute a minimum cost edge set $J \subseteq \hat {E}$ such that $G \cup J$ has connectivity k + 1. For the k-T-Connectivity Augmentation problem when $\hat {E}$ is an edge set on T we obtain ratio $O\left (\ln \frac {|T|}{|T|-k}\right )$, improving the ratio $O\left (\frac {|T|}{|T|-k} \cdot \ln \frac {|T|}{|T|-k}\right )$ 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 $4\frac {2}{3}$, improving the previous best ratio 12 of Nutov (ACM Trans. Algorithms, 9(1), 1, 2014). For the special case when all the edges in $\hat E$ are incident to s, we give a polynomial time algorithm, improving the ratio $4\frac {17}{30}$ of Kortsarz and Nutov, (2015) and Nutov (Algorithmica, 63(1-2), 398–410, 2012) for this variant.