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The multiple exchange property for matroid bases has recently beengeneralized for valuated matroids and M$^{\natural}$-concave set functions.This paper establishes a stronger form of this multiple exchange property thatimposes a cardinality condition on the exchangeable subset. The stronger formimmediately implies the defining exchange property of M$^{\natural}$-concaveset functions, which was not the case with the recently established multipleexchange property without the cardinality condition.

A Stronger Multiple Exchange Property for M$^{\natural}$-concave Functions