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 Király (in: Cook, Lovász, Vygen (eds) Research trends in combinatorial optimization, Springer, Berlin, pp 87–126, 2009), if and only if it is a generalized polymatroid.