Approximating Minimum Cost Connectivity Problems via Uncrossable Bifamilies and Spider-Cover Decompositions

We give approximation algorithms for the Generalized Steiner Network (GSN) problem. The input consists of a graph G = (V, E) with edge/node costs, a node subset S ¿ V, and connectivity requirements {r(s, t) : s,t ¿ T ¿ V}. The goal is to find a minimum cost subgraph H that for all s, t ¿ T contains r(s, t) pairwise edge-disjoint si-paths so that no two of them have a node in S - {s, t} in common. Three extensively studied particular cases are: Edge-GSN (S = 0), Node-GSN (S = V), and Element-GSN (r(s,t) = 0 whenever s ¿ S or t ¿ S). Let k = max_s,t¿T r(s, t). In Rooted GSN there is s ¿ T so that r(u, t) = 0 for all u¿s, and in the Subset k-Connected Subgraph problem r(s, t) = k for all s, t ¿ T. For edge costs, our ratios are: O(k²) for Rooted GSN and O(k² log k) for Subset k-Connected Subgraph. This improves the previous ratio O(k² log n) and settles the approximability of these problems to a constant for bounded k. For node-cost, our ratios are: (1) O(k log |T|) for Element-GSN, matching the best known ratio for Edge-GSN. (2) O(k² log |T|) for Rooted GSN and O(k³ log |T|) for Subset k-Connected Subgraph, improving the ratio O(k^s log² |T|). (3) O(k⁴ log² |T|) for GSN; this is the first non-trivial approximation algorithm for the problem.