Skip to main content
SpringerLink
Log in
Book cover

International Conference on Integer Programming and Combinatorial Optimization

IPCO 2004: Integer Programming and Combinatorial Optimization pp 352–367Cite as

A Capacity Scaling Algorithm for M-convex Submodular Flow

  • Conference paper

  • 4142 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3064))

Abstract

This paper presents a faster algorithm for the M-convex submodular flow problem, which is a generalization of the minimum-cost flow problem with an M-convex cost function for the flow-boundary, where an M-convex function is a nonlinear nonseparable discrete convex function on integer points. The algorithm extends the capacity scaling approach for the submodular flow problem by Fleischer, Iwata and McCormick (2002) with the aid of a novel technique of changing the potential by solving maximum submodular flow problems.

Supported by the Kayamori Foundation of Informational Science Advancement.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cunningham, W.H., Frank, A.: A primal-dual algorithm for submodular flows. Math. Oper. Res. 10, 251–262 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93, 214–250 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Edmonds, J., Giles, R.: A min-max relation for submodular functions on graphs. Ann. Discrete Math. 1, 185–204 (1977)

    Article  MathSciNet  Google Scholar 

  4. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)

    Article  MATH  Google Scholar 

  5. Fleischer, L., Iwata, S., McCormick, S.T.: A faster capacity scaling algorithm for minimum cost submodular flow. Math. Program., Ser. A 92, 119–139 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Frank, A.: Finding feasible vectors of Edmonds-Giles Polyhedra. J. Comb. Theory, Ser. B 36, 221–239 (1984)

    Article  MATH  Google Scholar 

  7. Fujishige, S.: Submodular Functions and Optimization. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  8. Fujishige, S., Zhang, X.: New algorithms for the intersection problem of submodular systems. Japan J. Indust. Appl. Math. 9, 369–382 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Iwata, S.: A capacity scaling algorithm for convex cost submodular flows. Math. Program. 76, 299–308 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Iwata, S.: A faster scaling algorithm for minimizing submodular functions. SIAM J. Comput. 32, 833–840 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Iwata, S., Shigeno, M.: Conjugate scaling algorithm for Fenchel-type duality in discrete convex optimization. SIAM J. Optim. 13, 204–211 (2003)

    Article  MathSciNet  Google Scholar 

  13. Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming–The State of the Art, pp. 235–257. Springer, Heidelberg (1983)

    Google Scholar 

  14. Moriguchi, S., Murota, K.: Capacity scaling algorithm for scalable M-convex submodular flow problems. Optim. Methods Softw. 18, 207–218 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Murota, K.: Valuated matroid intersection, I: optimality criteria. SIAM J. Discrete Math. 9, 545–561 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Murota, K.: Submodular flow problem with a nonseparable cost function. Combinatorica 19, 87–109 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  18. Murota, K., Tamura, A.: Application of M-convex submodular flow problem to mathematical economics. Japan J. Indust. Appl. Math. 20, 257–277 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B 80, 346–355 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Shioura, A.: Fast scaling algorithms for M-convex function minimization with application to resource allocation problem. Discrete Appl. Math. 134, 303–316 (2003)

    Article  MathSciNet  Google Scholar 

  21. Tamura, A.: Coordinatewise domain scaling algorithm for M-convex function minimization. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 21–35. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Tokyo, Tokyo, 113-8656, Japan

    Satoru Iwata

  2. JST, Saitama, 332-0012, Japan

    Satoko Moriguchi

  3. University of Tokyo, JST, Saitama, Tokyo, 113-8656, PRESTO, Japan

    Kazuo Murota

Authors
  1. Satoru Iwata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Satoko Moriguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kazuo Murota
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of IEOR, Columbia University, 500 West 120th Street, 10027, New York, NY, USA

    Daniel Bienstock

  2. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    George Nemhauser

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Iwata, S., Moriguchi, S., Murota, K. (2004). A Capacity Scaling Algorithm for M-convex Submodular Flow. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-25960-2_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22113-5

  • Online ISBN: 978-3-540-25960-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation