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The method of cutting the vertices of the permutation polyhedron graph is proposed for the solution of linear conditional optimization problems on permutations. The concepts of the permutation polyhedron graph and of partial graph of the set of permutations are introduced. The numerical experiments based on the created software implementation of the algorithm are described.

A direct cut-off method to solve combinatorial optimization problems on polyarrangements with additional constraints is proposed and justified. The method allows obtaining a feasible solution at each stage without constructing the linear hull of the set of polyarrangements.

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