Simpler exchange axioms for M-concave functions on generalized polymatroids

Abstract

$${\mathrm{M}}^\natural $$ -concave functions form a class of discrete concave functions in discrete convex analysis, and are defined by a certain exchange axiom. We show in this paper that $${\mathrm{M}}^\natural $$ -concave functions can be characterized by a combination of two simpler exchange properties. It is also shown that for a function defined on an integral polymatroid, a much simpler exchange axiom characterizes $${\mathrm{M}}^\natural $$ -concavity. These results have some significant implications in discrete convex analysis.