A characterization of a cone of pseudo-Boolean functions via supermodularity-type inequalities

A pseudo-Boolean function is a real valued function defined on the vertices of the unit n-dimensional hypercube. It has a unique expression as a multilinear polynomial in n variables. It is called almost-positive if all the coefficients in that expression, except maybe those in the linear part, are nonnegative. The almost-positive functions form a convex cone, given explicitly by its extreme rays. Here we describe this cone by a system of linear inequalities, which can be viewed as a natural generalization of supermodularity to higher orders. We also point out a characterization in terms of the sign of partial derivatives.