Approximating survivable networks with minimum number of steiner points

Given a graph H = (U, E) and connectivity requirements r = {r(u,v): u, v ∈ R ⊆ U}, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, ∥·∥) and connectivity requirements, find a minimum size set S ⊂ M \ V of additional points, such that the unit disc graph induced by U = V S satisfies the requirements. In the (node-connectivity) Survivable Network Design Problem (SNDP) we are given a graph G = (V, E) with edge costs and connectivity requirements, and seek a minimum cost subgraph H of G that satisfies the requirements. Let k = max _u,v∈V r(u, v) denote the maximum connectivity requirement. We will show a natural transformation of an SN-MSP instance (V, r) into an SNDP instance (G = (V, E), c, r), such that an α-approximation algorithm for the SNDP instance implies an α · O(k ²)-approximation algorithm for the SN-MSP instance. In particular, for the case of uniform requirements r(u, v) = k for all u, v ∈ V, we obtain for SN-MSP the ratio O(k ² ln k), which solves an open problem from (Bredin et al. Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) (2005), 309-319).