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

  • Kazuo MurotaEmail author
Original Paper Area 2


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.


Discrete convex analysis Matroid Exchange property Combinatorial optimization 

Mathematics Subject Classification

52B40 52A41 



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.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.School of Business AdministrationTokyo Metropolitan UniversityTokyoJapan

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