Approximating survivable networks with minimum number of steiner points

AbstractGiven 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 = maxu,v ∈ Vr(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(k2)‐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(k2 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). © 2012 Wiley Periodicals, Inc. NETWORKS, 2012