On combinatorial descriptions of faces of the cone of supermodular functions

Abstract

Five different ways of combinatorial description of non-empty faces of the cone of supermodular functions on the power set of a finite basic set $N$ are introduced. Their identification with faces of the cone of supermodular games allows one to associate to them certain polytopes in $\mathbb{R}^{N}$, known as cores (of these games) in context of cooperative game theory, or generalized permutohedra in context of polyhedral geometry. Non-empty faces of the supermodular cone then correspond to normal fans of those polytopes. This (basically) geometric way of description of faces of the cone then leads to the combinatorial ways of their description. The first combinatorial way is to identify the faces with certain partitions of the set of enumerations of $N$, known as rank tests in context of algebraic statistics. The second combinatorial way is to identify faces with certain collections of posets on $N$, known as (complete) fans of posets in context of polyhedral geometry. The third combinatorial way is to identify the faces with certain coverings of the power set of $N$, introduced relatively recently in context of cooperative game theory under name core structures. The fourth combinatorial way is to identify the faces with certain formal conditional independence structures, introduced formerly in context of multivariate statistics under name structural semi-graphoids. The fifth way is to identify the faces with certain subgraphs of the permutohedral graph, whose nodes are enumerations of $N$. We prove the equivalence of those six ways of description of non-empty faces of the supermodular cone. This result also allows one to describe the faces of the polyhedral cone of (rank functions of) polymatroids over $N$ and the faces of the submodular cone over $N$.