Matroid rank functions and discrete concavity

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.