Approximating minimum-cost connectivity problems via uncrossable bifamilies

• Authors: Zeev Nutov
• Venue: ACM Transactions on Algorithms (TALG)
• Year: 2012

We give approximation algorithms for the Survivable Network 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 of G that for all s,t ∈ T contains r(s,t) pairwise edge-disjoint st-paths such that no two of them have a node in S ∖ {s,t} in common. Three extensively studied particular cases are: Edge-Connectivity Survivable Network (S = ∅), Node-Connectivity Survivable Network (S = V), and Element-Connectivity Survivable Network (r(s,t) = 0 whenever s ∈ S or t ∈ S). Let k = maxs,t ∈ T r(s,t). In Rooted Survivable Network, there is s ∈ T such 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(k log k) for Rooted Survivable Network and O(k2 log k) for Subset k-Connected Subgraph. This improves the previous ratio O(k2 log n), and for constant values of k settles the approximability of these problems to a constant. For node-costs, our ratios are as follows. —O(k log |T|) for Element-Connectivity Survivable Network, matching the best known ratio for Edge-Connectivity Survivable Network. —O(k2 log |T|) for Rooted Survivable Network and O(k3 log |T|) for Subset k-Connected Subgraph, improving the ratio O(k8 log2 |T|). —O(k4 log2 |T|) for Survivable Network; this is the first nontrivial approximation algorithm for the node-costs version of the problem.