  The problem of determining the set of possible eigenvalues of 3 Hermitian
matrices that sum up to zero is known as the Horn problem. The answer is a
polyhedral cone, which, following Knutson and Tao, can be described as the
projection of a simpler cone in the space of triangular tableaux (or hives) to
the boundary nodes of the tableau.
  In this paper, we introduce a combinatorial problem defined in terms of
certain weighted planar graphs giving rise to exactly the same polyhedral cone.
In our framework, the values at the inner nodes of the triangular tableaux
receive a natural interpretation. Other problems of linear algebra fit into the
same scheme, among them the Gelfand--Zeitlin problem. Our approach is motivated
by the works of Fomin and Zelevinsky on total positivity and by the ideas of
tropicalization.
