Permutohedra and minimal matrices

The notions of minimality, π-uniqueness and additivity originated in discrete tomography. They have applications to Kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems. Here, we introduce the notion of real-minimality and give geometric characterizations of all these notions for a matrix A, by considering the intersection of the permutohedron determined by A with the transportation polytope in which A lies. We also study the computational complexity of deciding if the properties of being additive, real-minimal, π-unique and minimal hold for a given matrix, and show how to efficiently construct some matrix with any of these properties.