# Improved Approximation Algorithms for Min-Cost Connectivity Augmentation Problems

• Zeev Nutov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

## Abstract

A graph G is k-connected if it has k internally-disjoint st-paths for every pair st 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 $$(k+1)$$-(sT)-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.

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