Base polyhedra and the linking property

Abstract

An integer polyhedron $$P \subseteq {\mathbb {R}}^n$$ has the linking property if for any $$f \in {\mathbb {Z}}^n$$ and $$g \in {\mathbb {Z}}^n$$ with $$f \le g$$ , P has an integer point between f and g if and only if it has both an integer point above f and an integer point below g. We prove that an integer polyhedron in the hyperplane $$\sum _{j=1}^n x_j=\beta $$ is a base polyhedron if and only if it has the linking property. The result implies that an integer polyhedron has the strong linking property, as defined in Frank and Kiraly (in: Cook, Lovasz, Vygen (eds) Research trends in combinatorial optimization, Springer, Berlin, pp 87–126, 2009), if and only if it is a generalized polymatroid.