Tight approximation algorithm for connectivity augmentation problems

  • Authors:

  • Guy Kortsarz;Zeev Nutov

  • Affiliations:

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

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2008

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Abstract

The S-connectivity@l"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 G"0=(V,E"0), S@?V, and requirements r(u,v) on VxV, find a minimum size set F of new edges (any edge is allowed) so that @l"G"""0"+"F^S(u,v)=r(u,v) for all u,v@?V. Extensively studied particular choices of S are the edge-CA (when S=@A) 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 @W(2^l^o^g^^^1^^^-^^^@e^n). The only polylogarithmic approximation ratio known for CA was for rooted requirements-O(logn@?logr"m"a"x)=O(log^2n), where r"m"a"x=max"u","v"@?"Vr(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(logn) for SV arbitrary, and O(r"m"a"x@?logn) for S=V.