Improved Approximation Algorithms for Min-Cost Connectivity Augmentation Problems

Abstract

A graph G is k-connected if it has k internally-disjoint st-paths for every pair s, t of nodes. Given a root s and a set T of terminals is k-$(s,T)$-connected if it has k internally-disjoint st-paths for every $t \in T$. We consider two well studied min-cost connectivity augmentation problems, where we are given an integer $k \ge 0$, a graph $G=(V,E)$, and and an edge set F on V with costs. The goal is to compute a minimum cost edge set $J \subseteq F$ such that $G+J$ has connectivity $k+1$. In the k -Connectivity Augmentation problem G is k-connected and $G+J$ should be $(k+1)$-connected. In the $k$-$(s,T)$ -Connectivity Augmentation problem G is k-$(s,T)$-connected and $G+J$ should be $(k+1)$-(s, T)-connected.

For the k -Connectivity Augmentation problem we obtain the following results. For $n \ge 3k-5$, we obtain approximation ratios 3 for directed graphs and 4 for undirected graphs,improving the previous ratio 5 of [26]. 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 $2$-$(s,T)$ -Connectivity Augmentation problem we achieve ratio $4\frac{2}{3}$, improving the previous best ratio 12 of [24]. For the special case when all the edges in F are incident to s, we give a polynomial time algorithm, improving the ratio $4\frac{17}{30}$ of [21, 25] for this variant.