Abstract
Theorems about the characterization and exponential growth of the denominators of fractional components of noninteger vertices of the relaxation polyhedron in the multi-index axial assignment problem are proved. They made it possible to obtain new lower bounds on the number of noninteger vertices of this polyhedron and to refute the conjecture on the estimate of the ratio of the number of integer vertices to the number of all vertices of the multi-index axial transportation polyhedron determined by integer vectors.
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