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Approximating Node Connectivity Problems via Set Covers

  • Guy Kortsarz
  • Zeev Nutov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1913)

Abstract

We generalize and unify techniques from several papers to obtain relatively simple and general technique for designing approximation algorithms for finding min-cost k-node connected spanning subgraphs. For the general instance of the problem, the previously best known algorithm has approximation ratio 2k. For k ≤ 5, algorithms with approximation ratio [(k+1)/2] are known. For metric costs Khuller and Raghavachari gave a \( \left( {2 + \frac{{2(k - 1)}} {n}} \right) \)-approximation algorithm. We obtain the following results.

  1. (i)

    An I(k-k 0)-approximation algorithm for the problem of making a k0-connected graph k-connected by adding a minimum cost edge set, where \( I\left( k \right) = 2 + \sum\nolimits_{j = 1}^{\left\lfloor {\tfrac{k} {2}} \right\rfloor - 1} {\tfrac{1} {j}\left\lfloor {\tfrac{k} {{j + 1}}} \right\rfloor } \)

     
  2. (ii)

    A \( (2 + {\mathbf{ }}\frac{{k - 1}} {n}) \)-approximation algorithm for metric costs.

     
  3. (iv)

    A [(k + 1)/2]-approximation algorithm fork= 6, 7.

     
  4. (v)

    A fast [(k + 1)/2]-approximation algorithm fork= 4. The multiroot problem generalizes the min-cost k-connected subgraph problem. In the multiroot problem, requirements ku for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{ku, kv} internally disjoint paths between every pair of nodes u, v. For the general instance of the problem, the best known algorithm has approximation ratio 2k, wherek= maxku. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve fork= 7 the approximation guarantee from 3 to \( 2 + \frac{{\left\lfloor {\left( {k - 1} \right)/2} \right\rfloor }} {k} < 2.5 \)

     

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Guy Kortsarz
    • 1
  • Zeev Nutov
    • 1
  1. 1.Open University of IsraelRamat-AvivIsrael

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