Tight approximation algorithm for connectivity augmentation problems

The S-connectivityλ_G^S (u, v) of (u, v) in a graph G is the maximum number of uv-paths that no two of them have an edge or a node in S - {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G₀ = (V, E₀), S ⊆ V, and requirements r (u, v) on V × V, find a minimum size set F of new edges (any edge is allowed) so that λ_{G0 + F}^S (u, v) ≥ r (u, v) for all u, v ∈ V. Extensively studied particular choices of S are the edge-CA (when S = ∅) and the node-CA (when S = V). A. Frank gave a polynomial algorithm for undirected edge-CA and observed that the directed case even with rooted{0, 1} -requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V - S so that if r (u, v) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node-CA have approximation threshold Ω (2^{log1 - ε n}). The only polylogarithmic approximation ratio known for CA was for rooted requirements-O (log n ṡ log r_max) = O (log² n), where r_max = max_{u, v ∈ V} r (u, v). No nontrivial approximation algorithms were known for directed CA even for r (u, v) ∈ {0, 1}, nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O (log n) for S ≠ V arbitrary, and O (r_max ṡ log n) for S = V.