Sum of Sets of Integer Points of Common-Normal Faces of Integer Polyhedra

We consider the Minkowski sum of subsets of integer lattice, each of which is a set of integer points of a face of an extended submodular [Kashiwabara–Takabatake, Discrete Appl. Math. 131 (2003) 433] integer polyhedron supported by a common positive vector. We show a sufficient condition for the sum to contain all the integer points of its convex hull and a sufficient condition for the sum to include a specific subset of congruent integer points of its convex hull. The latter also gives rise to a subclass of extended submodular integer polyhedra, for which the sum of “copies” of a set of integer points of a face always contains all the integer points of its convex hull. We do this by using the properties of M-convex sets [Murota, Discrete Convex Analysis (2003)] and some logic from the elementary number theory. Our study has a direct significance for an economic problem of division of a bundle of indivisible goods.