Tight Approximation Algorithm for Connectivity Augmentation Problems

Abstract

The S-connectivity λ ^S _G (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 G ₀=(V,E ₀), 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 $\lambda^S_{G_0+F}(u,v) \geq r(u,v)$ for all u,v ∈V. Extensively studied particular cases 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 r(u,v) ∈{0,1} is at least as hard as the Set-Cover problem. Both directed and undirected node-CA have approximation threshold $\Omega(2^{\log^{1-\varepsilon}n})$. We give an approximation algorithm that matches these approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O(logn) for S ≠V arbitrary, and O(r _max logn) for S=V, where r _max= max _{u,v ∈ V} r(u,v).