Approximating survivable networks with β-metric costs

The Survivable Network Design (SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with -metric costs when c(xz) ≤ [c(xy) + c(yz)] for all x, y, z ∈ V , which varies from uniform costs ( = 1/2) to metric costs ( = 1). For the k-Connected Subgraph (k-CS) problem our ratios are: 1 + 2 k(1− ) − 1 2k−1 for undirected graphs, and 1 + 4 3 k(1−3 ) − 1 2k−1 for directed graphs and 12 ≤ < 1 3 . For undirected graphs this improves the ratios 1− of [2] and 2+ k n of [9] for all k ≥ 4 and 1 2 + 1 2k ≤ ≤ k 2 (k+1)−2 . We also show that SND admits the approximation ratios 2 1− for undirected graphs, and 4 3 1−3 2 for directed graphs and 1/2 ≤ ≤ 1/ √ 3. For two important particular cases of SND, so called Rooted SND and Subset k-CS, our ratios are 2 3 1−3 2 for directed graphs and 1− for Subset k-CS on undirected graphs.