In this thesis we study the relation between two problems of linear algebra concerning eigenvalues of Hermitian matrices and certain planar graphs: the Horn problem of describing the set of eigenvalues of sums of Hermitian matrices and the Gelfand-Zeitlin problem of describing the set of eigenvalues of Hermitian matrices and their principle submatrices. Both sets are polyhedral cones. We introduce a combinatorial framework where the same cones arise naturally. We also give some explaination of this unexpected relation and generalize our construction to arbitrary positive semirings.