Theory of Computing Systems

, Volume 62, Issue 3, pp 510–532 | Cite as

Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

  • Zeev NutovEmail author


Let κ G (s, t) denote the maximum number of pairwise internally disjoint st-paths in a graph G = (V, E). For a set \(T \subseteq V\) of terminals, G is k-T-connected if κ G (s, t) ≥ k for all s, tT; if T = V then G is k-connected. Given a root node s, G is k- (T, s)-connected if κ G (t, s) ≥ k for all tT. We consider the corresponding min-cost connectivity augmentation problems, where we are given a graph G = (V, E) of connectivity k, and an additional edge set \(\hat E\) on V with costs. The goal is to compute a minimum cost edge set \(J \subseteq \hat {E}\) such that \(G \cup J\) has connectivity k + 1. For the k-T-Connectivity Augmentation problem when \(\hat {E}\) is an edge set on T we obtain ratio \(O\left (\ln \frac {|T|}{|T|-k}\right )\), improving the ratio \(O\left (\frac {|T|}{|T|-k} \cdot \ln \frac {|T|}{|T|-k}\right )\) of Nutov (Combinatorica, 34(1), 95–114, 2014). For the k -Connectivity Augmentation problem we obtain the following approximation ratios. For n ≥ 3k − 5, we obtain ratio 3 for directed graphs and 4 for undirected graphs, improving the previous ratio 5 of Nutov (Combinatorica, 34(1), 95–114, 2014). For directed graphs and k = 1, or k = 2 and n odd, we further improve to 2.5 the previous ratios 3 and 4, respectively. For the undirected 2-(T, s)-Connectivity Augmentation problem we achieve ratio \(4\frac {2}{3}\), improving the previous best ratio 12 of Nutov (ACM Trans. Algorithms, 9(1), 1, 2014). For the special case when all the edges in \(\hat E\) are incident to s, we give a polynomial time algorithm, improving the ratio \(4\frac {17}{30}\) of Kortsarz and Nutov, (2015) and Nutov (Algorithmica, 63(1-2), 398–410, 2012) for this variant.


Node-connectivity augmentation Approximation algorithm Crossing biset family 


  1. 1.
    Aazami, A., Cheriyan, J., Laekhanukit, B.: A bad example for the iterative rounding method for mincost k-connected spanning subgraphs. Discret. Optim. 10 (1), 25–41 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auletta, V., Dinitz, Y., Nutov, Z., Parente, D.: A 2-approximation algorithm for finding an optimum 3-vertex-connected spanning subgraph. J. of Algorithms 32 (1), 21–30 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cheriyan, J., Laekhanukit, B.: Approximation algorithms for minimum-cost k- (S, T) connected digraphs. SIAM J. Discrete Math. 27(3), 1450–1481 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheriyan, J., Laekhanukit, B., Naves, G., Vetta, A.: Approximating rooted steiner networks. ACM Trans. Algorithms 11(2), 8:1–8:22 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheriyan, J., Végh, L.: Approximating minimum-cost k-node connected subgraphs via independence-free graphs. SIAM J. Computing 43(4), 1342–1362 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheriyan, J., Vempala, S., Vetta, A.: An approximation algorithm for the min-cost k-vertex connected subgraph. SIAM J. Computing 32(4), 1050–1055 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chuzhoy, J., Khanna, S.: An O(k 3 n)-approximation algorithm for vertex-connectivity survivable network design. Theory of Computing 8(1), 401–413 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dinitz, Y., Nutov, Z.: A 3-approximation algorithm for finding optimum 4,5-vertex-connected spanning subgraphs. J. of Algorithms 32(1), 31–40 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fackharoenphol, J., Laekhanukit, B.: An \({O}(\log ^{2} k)\)-approximation algorithm for the k-vertex connected subgraph problem. SIAM J. Computing 41, 1095–1109 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fleischer, L., Jain, K., Williamson, D.: Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. J. Comput. Syst. Sci. 72 (5), 838–867 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Frank, A.: Rooted k-connections in digraphs. Discret. Appl. Math. 157(6), 1242–1254 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Frank, A., Jordán, T.: Minimal edge-coverings of pairs of sets. J. of Comb. Theory B 65, 73–110 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frank, A., Tardos, E.: An application of submodular flows. Linear Algebra Appl. 114/115, 329–348 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fukunaga, T.: Approximating minimum cost source location problems with local vertex-connectivity demands. J. Discrete Algorithms 19, 30–38 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fukunaga, T.: Approximating the generalized terminal backup problem via half-integral multiflow relaxation. SIAM J. Discrete Math. 30(2), 777–800 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fukunaga, T., Nutov, Z., Ravi, R.: Iterative rounding approximation algorithms for degree-bounded node-connectivity network design. SIAM J. Computing 44(5), 1202–1229 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goemans, M., Goldberg, A., Plotkin, S., Shmoys, D., Tardos, E., Williamson, D.: Improved approximation algorithms for network design problems. In: SODA, pp. 223–232 (1994)Google Scholar
  18. 18.
    Grandoni, F., Laekhanukit, B.: Surviving in directed graphs: A polylogarithmic approximation for two-connected directed steiner tree. To appear in STOC’17Google Scholar
  19. 19.
    Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9, 256–278 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jordán, T.: On the optimal vertex-connectivity augmentation. J. on Comb. Theory B 63, 8–20 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khuller, S., Raghavachari, B.: Improved approximation algorithms for uniform connectivity problems. J. of Algorithms 21, 434–450 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kortsarz, G., Nutov, Z.: Approximating node connectivity problems via set covers. Algorithmica 37, 75–92 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kortsarz, G., Nutov, Z.: Approximating k-node connected subgraphs via critical graphs. SIAM J. on Computing 35(1), 247–257 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kortsarz, G., Nutov, Z.: Approximating source location and star survivable network problems. In: WG. To appear in Theoretical Computer Science, pp. 203–218 (2015)Google Scholar
  26. 26.
    Laekhanukit, B.: Parameters of two-prover-one-round game and the hardness of connectivity problems SODA, pp. 1626–1643 (2014)Google Scholar
  27. 27.
    Lando, Y., Nutov, Z.: Inapproximability of survivable networks. Theor. Comput. Sci. 410(21-23), 2122–2125 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nutov, Z.: Approximating minimum cost connectivity problems via uncrossable bifamilies. ACM Trans. Algorithms 9(1), 1 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nutov, Z.: Approximating node-connectivity augmentation problems. Algorithmica 63(1-2), 398–410 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nutov, Z.: Approximating subset k-connectivity problems. J. Discrete Algorithms 17, 51–59 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nutov, Z.: Approximating minimum-cost edge-covers of crossing biset families. Combinatorica 34(1), 95–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nutov, Z.: Improved approximation algorithms for min-cost connectivity augmentation problems. In: CSR, pp. 324–339 (2016)Google Scholar
  34. 34.
    Ravi, R., Williamson, D.P.: An approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica 18, 21–43 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ravi, R., Williamson, D.P.: Erratum: an approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica 34(1), 98–107 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Végh, L.: Augmenting undirected node-connectivity by one. SIAM J. Discrete Math. 25(2), 695–718 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.The Open University of IsraelRa’ananaIsrael

Personalised recommendations