Skip to main content
SpringerLink
Log in

Hardness of Approximation for Vertex-Connectivity Network-Design Problems

  • Conference paper
  • First Online:
Approximation Algorithms for Combinatorial Optimization (APPROX 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2462))

Abstract

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(n polylog(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.

Supported in part by NSF grants CCR-9820951 and CCR-0121555 and DARPA cooperative agreement F30602-00-2-0601.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.

    Article  MATH  MathSciNet  Google Scholar 

  2. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Cheriyan, T. Jordán, and Z. Nutov. On rooted node-connectivity problems. Algorithmica, 30(3):353–375, 2001.

    MATH  MathSciNet  Google Scholar 

  5. 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. J. Cheriyan and R. Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput., 30(2):528–560, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  7. 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. R. Diestel. Graph theory. Springer-Verlag, New York, second edition, 2000.

    Google Scholar 

  9. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. K. P. Eswaran and R. E. Tarjan. Augmentation problems. SIAM J. Comput., 5(4):653–665, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  11. U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105–124, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  13. U. Feige, M. M. Halldórsson, G. Kortsarz, and A. Srinivasan. Approximating the domatic number. SIAM J. Comput., 2002. To appear.

    Google Scholar 

  14. G. N. Frederickson and J. JáJá. Approximation algorithms for several graph augmentation problems. SIAM J. Comput., 10(2):270–283, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  15. 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. 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. A. Frank and É. Tardos. An application of submodular flows. Linear Algebra Appl., 114/115:329–348, 1989.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. T. Jordán. On the optimal vertex-connectivity augmentation. J. Combin. Theory Ser. B, 63(1):8–20, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  22. 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. 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. G. Kortsarz. On the hardness of approximating spanners. Algorithmica, 30(3):432–450, 2001.

    MATH  MathSciNet  Google Scholar 

  25. S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434–450, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Khuller and R. Thurimella. Approximation algorithms for graph augmentation. J. Algorithms, 14(2):214–225, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Petrank. The hardness of approximation: gap location. Comput. Complexity, 4(2):133–157, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  28. C. H. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two. Math. Oper. Res., 18(1):1–11, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763–803, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Ravi and D. P. Williamson. An approximation algorithm for minimumcost vertex-connectivity problems. Algorithmica, 18(1):21–43, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  31. 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. V. V. Vazirani. Approximation algorithms. Springer-Verlag, Berlin, 2001.

    Google Scholar 

  33. T. Watanabe and A. Nakamura. A minimum 3-connectivity augmentation of a graph. J. Comput. System Sci., 46(1):91–128, 1993.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Sciences, Rutgers University, 08102, Camden, NJ, USA

    Guy Kortsarz

  2. International Computer Science Institute (ICSI) and Computer Science Division, University of California, 94720, Berkeley, CA, USA

    Robert Krauthgamer

  3. Computer Science Division, University of California, 94720, Berkeley, CA, USA

    James R. Lee

Authors
  1. Guy Kortsarz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Krauthgamer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James R. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik und praktische Mathematik, Universität Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  2. Dipartimento di Informatika e Sistemistica, Universita di Roma La Sapienza, Via Salaria 113, 00198, Roma, Italy

    Stefano Leonardi

  3. College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, 30332-0280, Georgia, USA

    Vijay Vazirani

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kortsarz, G., Krauthgamer, R., Lee, J.R. (2002). Hardness of Approximation for Vertex-Connectivity Network-Design Problems. In: Jansen, K., Leonardi, S., Vazirani, V. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2002. Lecture Notes in Computer Science, vol 2462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45753-4_17

Download citation

  • DOI: https://doi.org/10.1007/3-540-45753-4_17

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44186-1

  • Online ISBN: 978-3-540-45753-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation