Approximation Algorithms for Connectivity Augmentation Problems

Abstract

In Connectivity Augmentation problems we are given a graph $$H=(V,E_H)$$ and an edge set E on V, and seek a min-size edge set $$J \subseteq E$$ such that $$H \cup J$$ has larger edge/node connectivity than H. In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by 1. In the Block-Tree Augmentation problem H is connected and $$H \cup S$$ should be 2-connected. In Leaf-to-Leaf Connectivity Augmentation problems every edge in E connects minimal deficient sets. For this version we give a simple combinatorial approximation algorithm with ratio 5/3, improving the 1.91 approximation of [6] (see also [23]), that applies for the general case. We also show by a simple proof that if the Steiner Tree problem admits approximation ratio $$\alpha $$ then the general version admits approximation ratio $$1+\ln (4-x)+\epsilon $$ , where x is the solution to the equation $$1+\ln (4-x)=\alpha +(\alpha -1)x$$ . For the currently best value of $$\alpha =\ln 4+\epsilon $$ [7] this gives ratio 1.942. This is slightly worse than the ratio 1.91 of [6], but has the advantage of using Steiner Tree approximation as a “black box”. In the Element Connectivity Augmentation problem we are given a graph $$G=(V,E)$$ , $$S \subseteq V$$ , and connectivity requirements $$r=\{r(u,v):u,v \in S\}$$ . The goal is to find a min-size set J of new edges on S such that for all $$u,v \in S$$ the graph $$G \cup J$$ contains r(u, v) uv-paths such that no two of them have an edge or a node in $$V \setminus S$$ in common. The problem is NP-hard even when $$\displaystyle r_{\max } = \max _{u,v \in S} r(u,v)=2$$ . We obtain ratio 3/2, improving the previous ratio 7/4 of [22]. For the case of degree bounds on S we obtain the same ratio with just $$+1$$ degree violation, which is tight, since deciding whether there exists a feasible solution is NP-hard even when $$r_{\max }=2$$ .