A modification of the combinatorial truncation method for optimization over vertex-located sets is considered. The modification allows dealing with degenerate solutions of auxiliary problems. The form of the truncating inequality is substantiated. An example is given to illustrate the application of the method.

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.

A conditional linear fully combinatorial minimization problem on permutations is analyzed. The methods of branching, pruning, and estimating in the branch and bound method are proposed for this problem. An illustrative example of applying the method to the problem is presented. The property of the proposed estimation of the admissible subset, which increases the efficiency of branching and pruning, is proved.