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

TL;DR

This paper introduces a stronger multiple exchange property for M$^{ atural}$-concave functions, adding a cardinality condition that enhances the understanding of their exchange properties.

Contribution

It establishes a stronger form of the multiple exchange property for M$^{ atural}$-concave functions, which implies the fundamental exchange property.

Findings

Stronger exchange property with cardinality condition

Immediate implication of the fundamental exchange property

Enhanced understanding of M$^{ atural}$-concave functions

Abstract

The multiple exchange property for matroid bases has recently been generalized for valuated matroids and 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 M$^{\natural}$-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.