Skip to main content
SpringerLink
Log in

Matroid rank functions and discrete concavity

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We discuss the relationship between matroid rank functions and a concept of discrete concavity called \({{\rm M}^{\natural}}\)-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are \({{\rm M}^{\natural}}\)-concave functions, while the (weighted) sum of matroid rank functions is not \({{\rm M}^{\natural}}\)-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions to be an \({{\rm M}^{\natural}}\)-concave function, and show that every weighted rank function can be represented as a weighted sum of matroid rank functions satisfying this condition.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint. In: Integer Programming and Combinatorial Optimization, 12th International IPCO Conference. Lecture Notes in Computer Science, vol. 4513, pp. 182–196. Springer, Berlin (2007)

  2. Dughmi, S.: A truthful randomized mechanism for combinatorial public projects via convex optimization. In: Proceedings of the 12th ACM Conference on Electronic Commerce (EC-2011), pp. 263–272. ACM, New York (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Dughmi, S., Roughgarden, T., Sundararajan, M.: Revenue submodularity. In: Proceedings of the 10th ACM Conference on Electronic Commerce (EC-2009), pp. 243–252. ACM, New York (2009)

  5. Dughmi, S., Roughgarden, T., Yan, Q.: From convex optimization to randomized mechanisms: toward optimal combinatorial auctions. In: Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, pp. 149–158. ACM, New York (2011)

  6. Dughmi, S., Vondrák, J.: Limitations of randomized mechanisms for combinatorial auctions. In: Proceedings of IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 502–511. IEEE, Los Alamitos (2011)

  7. Frank A., Frank A.: Generalized polymatroids and submodular flows. Math. Progr. 42, 489–563 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fujishige S.: Submodular Functions and Optimization. 2nd edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  9. Fujishige S., Yang S.: A note on Kelso and Crawford’s gross substitutes condition. Math. Oper. Res. 28, 463–469 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Iwata S., Murota K., Shigeno M.: A fast submodular intersection algorithm for strong map sequences. Math. Oper. Res. 22, 803–813 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lehmann B., Lehmann D.J., Nisan N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55, 270–296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Murota K.: Discrete convex analysis. Math. Progr. 83, 313–371 (1998)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Murota K.: Submodular function minimization and maximization in discrete convex analysis. RIMS Kokyuroku Bessatsu B23, 193–211 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Murota K., Shioura A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Oxley J.: Matroid Theory. Oxford University Press, New York (1992)

    MATH  Google Scholar 

  17. Reijnierse H., van Gellekom A., Potters J.A.M.: Verifying gross substitutability. Econ. Theory 20, 767–776 (2002)

    Article  MATH  Google Scholar 

  18. Schrijver A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

  19. Shioura A.: On the pipage rounding algorithm for submodular function maximization: a view from discrete convex analysis. Discret. Math. Algorithms Appl. 1, 1–23 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tardos, É.:Generalized matroids and supermodular colourings. In: Recski A.,Lovász L. (eds.) Matroid Theory, Colloquia Mathematica Societatis János Bolyai 40, pp. 359–382. North-Holland, Amsterdam (1985)

  21. Welsh D.J.A.: Matroid Theory. Academic Press, London (1976)

    MATH  Google Scholar 

  22. Welsh D.J.A. (1995) Matroids: fundamental concepts. In: Graham R.L., Grötschel M., Lovász L. (eds.) Handbook of Combinatorics, pp. 481–526. Elsevier, Amsterdam.

Download references

Author information

Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Akiyoshi Shioura

Authors
  1. Akiyoshi Shioura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akiyoshi Shioura.

Additional information

This work is partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

About this article

Cite this article

Shioura, A. Matroid rank functions and discrete concavity. Japan J. Indust. Appl. Math. 29, 535–546 (2012). https://doi.org/10.1007/s13160-012-0082-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-012-0082-0

Keywords

Mathematics cSubject Classification

Search

Navigation