Skip to main content
SpringerLink
Log in

On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

Abstract

We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A∈ℝm×n, b∈ℝm and a polytope P \( \subseteq\) ℝn, the fractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ∈-approximate solution to this problem is an x ∈ P such that Ax ≤ (1+∈)b. An ∈-relaxed decision procedure always finds an ∈-approximate solution if an exact solution exists.

Research supported by NSF NSF Grant CCR-9700146.

Research supported by NSF Career award CCR-9720664.

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. B. Awerbuch and T. Leighton. A simple local-control approximation algorithm for multicommodity flow. In Proc. of the 34th IEEE Annual Symp. on Foundation of Computer Science, pages 459–468, 1993.

    Google Scholar 

  2. B. Awerbuch and T. Leighton. Improved approximation algorithms for the multicommodity flow problem and local competitive routing in dynamic networks. In Proc. of the 26th Ann. ACM Symp. on Theory of Computing, pages 487–495, 1994.

    Google Scholar 

  3. J. F. Benders Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4:238–252, 1962.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. B. Dantzig and P. Wolfe. Decomposition principle for linear programs. Operations Res., 8:101–111, 1960.

    MATH  Google Scholar 

  5. Y. Freund and R. Schapire. Adaptive game playing using multiplicative weights. J. Games and Economic Behavior, to appear.

    Google Scholar 

  6. N. Garg and J. Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In Proc. of the 39th Annual Symp. on Foundations of Computer Science, pages 300–309, 1998.

    Google Scholar 

  7. A. V. Goldberg. A natural randomization strategy for multicommodity flow and related problems. Information Processing Letters, 42:249–256, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. D. Grigoriadis and L. G. Khachiyan. A sublinear-time randomized approximation algorithm for matrix games. Technical Report LCSR-TR-222, Rutgers University Computer Science Department, New Brunswick, NJ, April 1994.

    Google Scholar 

  9. M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees. Operations Research, 18:1138–1162, 1971

    MathSciNet  Google Scholar 

  10. M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees: Part II. Mathematical Programming, 1:6–25, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Karger and S. Plotkin. Adding multiple cost constraints to combinatorial optimization problems, with applications to multicommodity flows. In Proc. of the 27th Ann. ACM Symp. on Theory of Computing, pages 18–25, 1995.

    Google Scholar 

  12. L. G. Khachiyan. Convergence rate of the game processes for solving matrix games. Zh. Vychisl. Mat. and Mat. Fiz., 17:1421–1431, 1977. English translation in USSR Comput. Math and Math. Phys., 17:78–88, 1978.

    MATH  Google Scholar 

  13. P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466–487, June 1994.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. R. Ford and D. R. Fulkerson. A suggested computation for maximal multicommodity network flow. Management Sci., 5:97–101, 1958.

    MATH  MathSciNet  Google Scholar 

  15. T. Leighton, F. Makedon, S. Plotkin, C. Stein, É. Tardos, and S. Tragoudas. Fast approximation algorithms for multicommodity flow problems. Journal of Computer and System Sciences, 50(2):228, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Plotkin, D. Shmoys, and É. Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20:257–301, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Radzik. Fast deterministic approximation for the multicommodity flow problem. In Proc. of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 486–492, 1995.

    Google Scholar 

  18. F. Shahrokhi and D. W. Matula. The maximum concurrent flow problem. JACM, 37:318–334, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. F. Shapiro. A survey of lagrangean techniques for discrete optimization. Annals of Discrete Mathematics, 5:113–138, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  20. N. E. Young. Randomized rounding without solving the linear program. In Proc. of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 170–178, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brown University, Providence, RI, USA

    Philip Klein

  2. Dartmouth College, Hanover, NH, USA

    Neal Young

Authors
  1. Philip Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neal Young
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Klein, P., Young, N. (1999). On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_24

Download citation

  • DOI: https://doi.org/10.1007/3-540-48777-8_24

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation