Approximation Algorithms for Connectivity Augmentation Problems

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⊆ E such that H∪ 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∪ 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 α then the general version admits approximation ratio 1 + ln (4 - x) + ϵ, where x is the solution to the equation 1 + ln (4 - x) = α+ (α- 1 ) x. For the currently best value of α= ln 4 + ϵ [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⊆ V, and connectivity requirements r= { r(u, v): u, v∈ S}. The goal is to find a min-size set J of new edges on S such that for all u, v∈ S the graph G∪ J contains r(u, v) uv-paths such that no two of them have an edge or a node in V\ S in common. The problem is NP-hard even when rmax=maxu,v∈Sr(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.