Skip to main content
SpringerLink
Log in

A stronger multiple exchange property for \(\hbox {M}^{\natural }\)-concave functions

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

The multiple exchange property for matroid bases has recently been generalized for valuated matroids and \(\hbox {M}^{\natural }\)-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of \(\hbox {M}^{\natural }\)-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality 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

Notes

  1. The assumption \(\mathrm{dom\,}g_{1} \cap \mathrm{dom\,}g_{2} \not = \emptyset \) in [8, Theorem 8.21 (1)] is satisfied, since \(\mathrm{dom\,}g_{1} = \mathrm{dom\,}g_{2} = {\mathbb {R}}^{N}\).

  2. (3.17) means a kind of strong quotient relation.

References

  1. Dress, A.W.M., Wenzel, W.: Valuated matroid: a new look at the greedy algorithm. Appl. Math. Lett. 3, 33–35 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Fujishige, S.: Submodular Functions and Optimization. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  4. Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87, 95–124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kelso Jr., A.S., Crawford, V.P.: Job matching, coalition formation, and gross substitutes. Econometrica 50, 1483–1504 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kung, J.P.S.: Basis-exchange properties. In: White, N. (ed.) Theory of Matroids, Chapter 4, pp. 62–75. Cambridge University Press, London (1986)

    Chapter  Google Scholar 

  7. Murota, K.: Fenchel-type duality for matroid valuations. Math. Program. 82, 357–375 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  MATH  Google Scholar 

  9. Murota, K.: Recent developments in discrete convex analysis. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, Chapter 11, pp. 219–260. Springer, Berlin (2009)

    Chapter  Google Scholar 

  10. Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Des. 1, 151–273 (2016)

    Google Scholar 

  11. Murota, K.: Multiple exchange property for M\(^{\natural }\)-concave functions and valuated matroids. Math. Oper. Res. (to appear)

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  14. Shioura, A., Tamura, A.: Gross substitutes condition and discrete concavity for multi-unit valuations: a survey. J. Oper. Res. Soc. Jpn. 58, 61–103 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tamura, A.: Applications of discrete convex analysis to mathematical economics. Publ. Res. Inst. Math. Sci. 40, 1015–1037 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tamura, A.: Discrete Convex Analysis and Game Theory (in Japanese). Asakura Publishing Co., Tokyo (2009)

    Google Scholar 

Download references

Acknowledgements

The author thanks Akiyoshi Shioura for suggesting a simplification in the proof of Sect. 3. He is also thankful to Kenjiro Takazawa and Akihisa Tamura for helpful comments.

Author information

Authors and Affiliations

  1. School of Business Administration, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Kazuo Murota

Authors
  1. Kazuo Murota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuo Murota.

Additional information

This work was supported by The Mitsubishi Foundation, CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Number 26280004.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Murota, K. A stronger multiple exchange property for \(\hbox {M}^{\natural }\)-concave functions. Japan J. Indust. Appl. Math. 35, 411–421 (2018). https://doi.org/10.1007/s13160-017-0278-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-017-0278-4

Keywords

Mathematics Subject Classification

Search

Navigation