Approximating Subset k-Connectivity Problems

  • Zeev Nutov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7164)


A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T; T is k-connected in J if T is k-connected to every s ∈ T. We consider the Subset k -Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, a node subset T ⊆ V, and a subgraph J = (V,E J ) of G such that T is (k − 1)-connected in J, find a minimum-cost augmenting edge-set F ⊆ E ∖ E J such that T is k-connected in J ∪ F. The problem admits trivial ratio O(|T|2). We consider the case |T| > k and prove that for directed/undirected graphs and edge/node-costs, a ρ-approximation algorithm for Rooted Subset k -Connectivity Augmentation implies the following approximation ratios for Subset k -Connectivity Augmentation:

(i) \(b(\rho+k) + {\left(\frac{|T|}{|T|-k}\right)}^2 O\left(\log \frac{|T|}{|T|-k}\right)\) and

(ii) \(\rho \cdot O\left(\frac{|T|}{|T|-k} \log k \right)\),

where b = 1 for undirected graphs and b = 2 for directed graphs. The best known values of ρ on undirected graphs are min {|T|,O(k)} for edge-costs and min {|T|,O(k log|T|)} for node-costs; for directed graphs ρ = |T| for both versions. Our results imply that unless k = |T| − o(|T|), Subset k -Connectivity Augmentation admits the same ratios as the best known ones for the rooted version. This improves the ratios in [19,14].


Directed Graph Undirected Graph Approximation Ratio Survivable Network Connectivity Requirement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Auletta, V., Dinitz, Y., Nutov, Z., Parente, D.: A 2-approximation algorithm for finding an optimum 3-vertex-connected spanning subgraph. J. Algorithms 32(1), 21–30 (1999)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cheriyan, J., Vetta, A.: Approximation algorithms for network design with metric costs. SIAM J. Discrete Mathematics 21(3), 612–636 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chuzhoy, J., Khanna, S.: An O(k 3 logn)-approximation algorithm for vertex-connectivity survivable network design. In: FOCS, pp. 437–441 (2009)Google Scholar
  4. 4.
    Dinitz, Y., Nutov, Z.: A 3-approximation algorithm for finding optimum 4,5-vertex-connected spanning subgraphs. J. Algorithms 32(1), 31–40 (1999)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Frank, A., Jordán, T.: Minimal edge-coverings of pairs of sets. J. Combinatorial Theory, Ser. B 65(1), 73–110 (1995)MATHCrossRefGoogle Scholar
  6. 6.
    Frank, A., Tardos, E.: An application of submodular flows. Linear Algebra and its Applications 114/115, 329–348 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jordán, T.: On the optimal vertex-connectivity augmentation. J. Combinatorial Theory, Ser. B 63(1), 8–20 (1995)MATHCrossRefGoogle Scholar
  8. 8.
    Khuller, S., Raghavachari, B.: Improved approximation algorithms for uniform connectivity problems. J. Algorithms 21(2), 434–450 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kortsarz, G., Krauthgamer, R., Lee, J.: Hardness of approximation for vertex-connectivity network design problems. SIAM J. Computing 33(3), 704–720 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kortsarz, G., Nutov, Z.: Approximating node-connectivity problems via set covers. Algorithmica 37, 75–92 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kortsarz, G., Nutov, Z.: Approximating k-node connected subgraphs via critical graphs. SIAM J. on Computing 35(1), 247–257 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kortsarz, G., Nutov, Z.: Approximating minimum-cost connectivity problems. In: Gonzalez, T.F. (ed.) Approximation algorithms and Metaheuristics, Ch. 58. Chapman & Hall/CRC (2007)Google Scholar
  13. 13.
    Kortsarz, G., Nutov, Z.: Tight approximation algorithm for connectivity augmentation problems. J. Computer and System Sciences 74(5), 662–670 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Laekhanukit, B.: An Improved Approximation Algorithm for Minimum-Cost Subset k-Connectivity. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 13–24. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Lando, Y., Nutov, Z.: Inapproximability of survivable networks. Theoretical Computer Science 410(21-23), 2122–2125 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Nutov, Z.: Approximating minimum-cost edge-covers of crossing biset families. In: Manuscript. Preliminary version: An almost O(logk)-approximation for k-connected subgraphs, SODA 2009, pp. 912–921 (2009)Google Scholar
  18. 18.
    Nutov, Z.: Approximating rooted connectivity augmentation problems. Algorithmica 44, 213–231 (2006)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Nutov, Z.: Approximating minimum cost connectivity problems via uncrossable bifamilies and spider-cover decompositions. In: FOCS, pp. 417–426 (2009)Google Scholar
  20. 20.
    Nutov, Z.: Approximating Node-Connectivity Augmentation Problems. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 286–297. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Végh, L.: Augmenting undirected node-connectivity by one. SIAM J. Discrete Mathematics 25(2), 695–718 (2011)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Zeev Nutov
    • 1
  1. 1.The Open University of IsraelIsrael

Personalised recommendations