Tight approximation algorithm for connectivity augmentation problems

  • Authors:

  • Guy Kortsarz;Zeev Nutov

  • Affiliations:

  • Rutgers University, Camden, NJ;The Open University of Israel, Raanana, Israel

  • Venue:
  • ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
  • Year:
  • 2006

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Abstract

The S-connectivityλSG(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 $\lambda^S_{G_0+F}(u,v) \geq r(u,v)$ 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(rmaxlogn) for S=V, where rmax= max u,v∈Vr(u,v).