Hardness of Approximation for Vertex-Connectivity Network-Design Problems

  • Guy Kortsarz
  • Robert Krauthgamer
  • James R. Lee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2462)


In the survivable network design problem SNDP, the goal is to find a minimum-cost subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.

We give the first lower bound on the approximability of SNDP, showing that the problem admits no efficient \( 2^{\log ^{1 - \in } n} \) ratio approximation for any fixed ∈>0 unless NP ⊆ DTIME(npolylog(n)). We also show hardness of approximation results for several important special cases of SNDP, including constant factor hardness for the k-vertex connected spanning subgraph problem (k-VCSS) and for the vertex-connectivity augmentation problem, even when the edge costs are severely restricted.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ADNP99.
    V. Auletta, Y. Dinitz, Z. Nutov, and D. Parente. A 2-approximation algorithm for finding an optimum 3-vertex-connected spanning subgraph. J. Algorithms, 32(1):21–30, 1999.MATHCrossRefMathSciNetGoogle Scholar
  2. AL96.
    S. Arora and C. Lund. Hardness of approximations. In D. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, 1996.Google Scholar
  3. ALM+98.
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501–555, 1998.MATHCrossRefMathSciNetGoogle Scholar
  4. CJN01.
    J. Cheriyan, T. Jordán, and Z. Nutov. On rooted node-connectivity problems. Algorithmica, 30(3):353–375, 2001.MATHMathSciNetGoogle Scholar
  5. CL99.
    A. Czumaj and A. Lingas. On approximability of the minimum-cost k-connected spanning subgraph problem. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 281–290. ACM, 1999.Google Scholar
  6. CT00.
    J. Cheriyan and R. Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput., 30(2):528–560, 2000.MATHCrossRefMathSciNetGoogle Scholar
  7. CVV02.
    J. Cheriyan, S. Vempala, and A. Vetta. Approximation algorithms for minimum-cost k-vertex connected subgraphs. In 34th Annual ACM Symposium on the Theory of Computing, 2002. To appear.Google Scholar
  8. Die00.
    R. Diestel. Graph theory. Springer-Verlag, New York, second edition, 2000.Google Scholar
  9. DN99.
    Y. Dinitz and Z. Nutov. A 3-approximation algorithm for finding optimum 4, 5-vertex-connected spanning subgraphs. J. Algorithms, 32(1):31–40, 1999.MATHCrossRefMathSciNetGoogle Scholar
  10. ET76.
    K. P. Eswaran and R. E. Tarjan. Augmentation problems. SIAM J. Comput., 5(4):653–665, 1976.MATHCrossRefMathSciNetGoogle Scholar
  11. Fei98.
    U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, 1998.MATHCrossRefMathSciNetGoogle Scholar
  12. Fer98.
    C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105–124, 1998.MATHCrossRefMathSciNetGoogle Scholar
  13. FHKS02.
    U. Feige, M. M. Halldórsson, G. Kortsarz, and A. Srinivasan. Approximating the domatic number. SIAM J. Comput., 2002. To appear.Google Scholar
  14. FJ81.
    G. N. Frederickson and J. JáJá. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270–283, 1981.MATHCrossRefMathSciNetGoogle Scholar
  15. FJW01.
    L. Fleischer, K. Jain, and D. P. Williamson. An iterative rounding 2-approximation algorithm for the element connectivity problem. In 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 339–347, 2001.Google Scholar
  16. Fra94.
    A. Frank. Connectivity augmentation problems in network design. In J. R. Birge and K. G. Murty, editors, Mathematical Programming: State of the Art 1994, pages 34–63. The University of Michigan, 1994.Google Scholar
  17. FT89.
    A. Frank and É. Tardos. An application of submodular flows. Linear Algebra Appl., 114/115:329–348, 1989.CrossRefMathSciNetGoogle Scholar
  18. HR91.
    T. Hsu and V. Ramachandran. A linear time algorithm for triconnectivity augmentation. In 32nd Annual IEEE Symposium on Foundations of Computer Science, pages 548–559, 1991.Google Scholar
  19. Hsu92.
    T. Hsu. On four-connecting a triconnected graph. In 33nd Annual IEEE Symposium on Foundations of Computer Science, pages 70–79, 1992.Google Scholar
  20. Jai01.
    K. Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21(1):39–60, 2001.MATHCrossRefMathSciNetGoogle Scholar
  21. Jor95.
    T. Jordán. On the optimal vertex-connectivity augmentation. J. Combin. Theory Ser. B, 63(1):8–20, 1995.MATHCrossRefMathSciNetGoogle Scholar
  22. Khu96.
    S. Khuller. Approximation algorithms for finding highly connected subgraphs. In D. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, 1996.Google Scholar
  23. KN00.
    G. Kortsarz and Z. Nutov. Approximating node connectivity problems via set covers. In 3rd International workshop on Approximation algorithms for combinatorial optimization (APPROX), pages 194–205. Springer, 2000.Google Scholar
  24. Kor01.
    G. Kortsarz. On the hardness of approximating spanners. Algorithmica, 30(3):432–450, 2001.MATHMathSciNetGoogle Scholar
  25. KR96.
    S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434–450, 1996.MATHCrossRefMathSciNetGoogle Scholar
  26. KT93.
    S. Khuller and R. Thurimella. Approximation algorithms for graph augmentation. J. Algorithms, 14(2):214–225, 1993.MATHCrossRefMathSciNetGoogle Scholar
  27. Pet94.
    E. Petrank. The hardness of approximation: gap location. Comput. Complexity, 4(2):133–157, 1994.MATHCrossRefMathSciNetGoogle Scholar
  28. PY93.
    C. H. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two. Math. Oper. Res., 18(1):1–11, 1993.MATHMathSciNetCrossRefGoogle Scholar
  29. Raz98.
    R. Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763–803, 1998.MATHCrossRefMathSciNetGoogle Scholar
  30. RW97.
    R. Ravi and D. P. Williamson. An approximation algorithm for minimumcost vertex-connectivity problems. Algorithmica, 18(1):21–43, 1997.MATHCrossRefMathSciNetGoogle Scholar
  31. RW02.
    R. Ravi and D. P. Williamson. Erratum: An approximation algorithm for minimum-cost vertex-connectivity problems. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1000–1001, 2002.Google Scholar
  32. Vaz01.
    V. V. Vazirani. Approximation algorithms. Springer-Verlag, Berlin, 2001.Google Scholar
  33. WN93.
    T. Watanabe and A. Nakamura. A minimum 3-connectivity augmentation of a graph. J. Comput. System Sci., 46(1):91–128, 1993.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Guy Kortsarz
    • 1
  • Robert Krauthgamer
    • 2
  • James R. Lee
    • 3
  1. 1.Department of Computer SciencesRutgers UniversityCamdenUSA
  2. 2.International Computer Science Institute (ICSI) and Computer Science DivisionUniversity of CaliforniaBerkeleyUSA
  3. 3.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA

Personalised recommendations