Cyclic Permutations in the Methods of Combinatorial Optimization on the Basis of Cyclic Transfers

Abstract

Permutations sets are very often considered in theoretical and applied research in the field of combinatorics andcombinatorial optimization. By now many properties of permutations have been investigated, in particular those associatedwith the cyclic structure of permutations. To solve optimization problems on combinatorial sets, a number of methodshave been developed and successfully used. Among them, there are methods for finding exact, approximate, and heuristicsolutions. Among the search methods for an approximate solution, search methods in the neighborhood are widely used.Among the combinatorial search methods in the neighborhood, it is necessary to distinguish a group of methods that usescyclic transfers. A general search scheme using cyclic transfers is known. It is based on the search for a cyclic transferin the auxiliary graph. But the features of the tasks being solved sometimes do not allow constructing an auxiliary graphand using the existing search methods. In such cases, an alternative approach is proposed. This is a general strategy forsolving problems using cyclic transfers, which is a modification of the strategy of Thompson and Orlin. We propose theconstruction of cyclic transfers based not on the use of an auxiliary graph, but based on the set of cyclic permutationsand its properties to obtain cyclic transfers. To demonstrate the strategy for solving combinatorial optimization problemsdescribed in the article, one of the vehicle routing problems was solved. Computational experiments were conducted todemonstrate the effectiveness of the proposed strategy. Their results are given in the article.