Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Approximation through local optimality: Designing networks with small degree

  • Conference paper
  • First Online:
Foundations of Software Technology and Theoretical Computer Science (FSTTCS 1992)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 652))

Abstract

We give quasipolynomial-time approximation algorithms for designing networks with minimum degree. Using our methods, one can design one-connected networks to meet a variety of connectivity requirements. The degree of the output network is guaranteed to be at most (1+ε) times optimal, plus an additive error of O(log n/ε) for any εs0. We also provide a quasipolynomial-time approximation algorithm for designing a two-edge-connected spanning subgraph of a given two-edge-connected graph of approximately minimum degree. The performance guarantee is identical to that for one-connected networks.

As a consequence of our analysis, we show that the minimum degree in both the problems above is well-estimated by certain polynomially solvable linear programs. This fact suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch-andbound.

Research supported by an IBM Graduate Fellowship. Additional support provided by NSF PYI award CCR-9157620 and DARPA contract N00014-91-J-4052 ARPA Order No. 8225.

Research supported by NSF grant CCR-9012357 and NSF PYI award CCR-9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Additional support provided by DARPA contract N00014-91-J-4052 ARPA Order No. 8225.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Agrawal, P. Klein and R. Ravi, “When trees collide: an approximation algorithm for the generalized Steiner tree problem on networks,” Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (1991), pp. 134–144.

    Google Scholar 

  2. A. Agrawal, P. Klein and R. Ravi, “How tough is the minimum-degree Steiner tree? An approximate min-max equality (complete with algorithms)”, TR-CS-91-49, Brown University (1991), Submitted to SIAM J. on Disc. Math.

    Google Scholar 

  3. D. Bauer, S.L. Hakimi, and E. Schmeichel, “Recognizing tough graphs is NP-hard,” Disc. Appl. Math. 28 (1990), pp. 191–195.

    Article  Google Scholar 

  4. V. Chvátal, “Tough graphs and Hamiltonian circuits,” Disc. Math. 5 (1973), pp. 215–228.

    Article  Google Scholar 

  5. J. Edmonds, and E. L. Johnson, “Matching, Euler tours and the Chinese postman”, Math. Prog. 5, (1973), pp. 88–124.

    Article  Google Scholar 

  6. M. Fürer and B. Raghavachari, “An NC approximation algorithm for the minimum-degree spanning tree problem,” Proceedings of the 28th Annual Allerton Conference on Communication, Control and Computing (1990), pp. 274–281.

    Google Scholar 

  7. M. Fürer and B. Raghavachari, “Approximating the minimum-degree spanning tree to within one from the optimal degree”, Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms (1992), pp. 317–324.

    Google Scholar 

  8. M. Fürer and B. Raghavachari, “Approximating the minimum-degree spanning and Steiner trees to within one from the optimal degree”, TR CS-92-13, Pennsylvania State University, June 1992.

    Google Scholar 

  9. M. R. Garey and D. S. Johnson, Computers and Intractability: A guide to the theory of NP-completeness, W. H. Freeman, San Francisco (1979).

    Google Scholar 

  10. M. X. Goemans, and D. P. Williamson, “A general approximation technique for constrained forest problems”, Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms (1992), pp. 307–316.

    Google Scholar 

  11. P. Klein and R. Ravi, “Approximation through uncrossing: From edge-cuts to node-cuts,” submitted to the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms.

    Google Scholar 

  12. G. L. Nemhauser, and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley Interscience series in Disc. Math. and Optimization (1988).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brown University, 02912, Providence, RI, USA

    R. Ravi & P. Klein

  2. Pennsylvania State University, University Park, 16802, PA, USA

    B. Raghavachari

Authors
  1. R. Ravi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Raghavachari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rudrapatna Shyamasundar

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ravi, R., Raghavachari, B., Klein, P. (1992). Approximation through local optimality: Designing networks with small degree. In: Shyamasundar, R. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1992. Lecture Notes in Computer Science, vol 652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56287-7_112

Download citation

  • DOI: https://doi.org/10.1007/3-540-56287-7_112

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56287-0

  • Online ISBN: 978-3-540-47507-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation