Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

Abstract

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.

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References

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Frank, A.: Rooted k-connections in digraphs. Discret. Appl. Math. 157(6), 1242–1254 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Frank, A., Jordán, T.: Minimal edge-coverings of pairs of sets. J. of Comb. Theory B 65, 73–110 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Frank, A., Tardos, E.: An application of submodular flows. Linear Algebra Appl. 114/115, 329–348 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Fukunaga, T.: Approximating minimum cost source location problems with local vertex-connectivity demands. J. Discrete Algorithms 19, 30–38 (2013)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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’17

  19. 19.

    Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9, 256–278 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Jordán, T.: On the optimal vertex-connectivity augmentation. J. on Comb. Theory B 63, 8–20 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Khuller, S., Raghavachari, B.: Improved approximation algorithms for uniform connectivity problems. J. of Algorithms 21, 434–450 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kortsarz, G., Nutov, Z.: Approximating node connectivity problems via set covers. Algorithmica 37, 75–92 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kortsarz, G., Nutov, Z.: Approximating k-node connected subgraphs via critical graphs. SIAM J. on Computing 35(1), 247–257 (2005)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Nutov, Z.: Approximating minimum cost connectivity problems via uncrossable bifamilies. ACM Trans. Algorithms 9(1), 1 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Nutov, Z.: Approximating node-connectivity augmentation problems. Algorithmica 63(1-2), 398–410 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Nutov, Z.: Approximating subset k-connectivity problems. J. Discrete Algorithms 17, 51–59 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Nutov, Z.: Approximating minimum-cost edge-covers of crossing biset families. Combinatorica 34(1), 95–114 (2014)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Ravi, R., Williamson, D.P.: Erratum: an approximation algorithm for minimum-cost vertex-connectivity problems. Algorithmica 34(1), 98–107 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Végh, L.: Augmenting undirected node-connectivity by one. SIAM J. Discrete Math. 25(2), 695–718 (2011)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Zeev Nutov.

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A preliminary version of this paper appeared in [33].

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Nutov, Z. Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems. Theory Comput Syst 62, 510–532 (2018). https://doi.org/10.1007/s00224-017-9786-5

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Keywords

  • Node-connectivity augmentation
  • Approximation algorithm
  • Crossing biset family