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,i¾źt 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,i¾źT-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.