Method of ordering the values of a linear function on a set of permutations

Abstract

The paper deals with a new method of solving a combinatorial problem with account for the properties of the set of permutations and its structure. Using this method, the values of the linear objective function are sequenced and the set of permutations is decomposed over hyperplanes, with account of element recurrences. This makes it possible to develop an algorithm of finding the point (an element of the set of permutations) at which the objective function attains a given value.

Modified Coordinate Method to Solve Multicriteria Optimization Problems on Combinatorial Configurations

“…The coordinate method of localization of the value of linear function, modified to find the points satisfying additional constraints of the problem, allows reducing the number of points under consideration [6,7]. This approach is based on the properties of points that determine elements of the combinatorial configurations decomposed into subgraphs according to the selected type of nodes and represented as a subgraph scheme [7] where elements are ordered, and the value of the function on a certain subgraph is between the values at extreme nodes of the scheme.…”

“…As the number of optimization criteria increases, the problem complexity grows [6][7][8][9][10][11] and the need arises to develop a new approach to the solution of vector combinatorial problems since the general algorithm can not always be adapted and applied efficiently. Therefore, the form of its mathematical formulation and search for solution methods are important.…”

Construction of Hamiltonian paths in graphs of permutation polyhedra

“…This work continues the investigation of combinatorial problems over various sets of permutations, combinations, and polypermutations considered in [4,5,[10][11][12][13]. In this article, based on the established interrelation between problems over combinatorial sets and graphs of polyhedra of corresponding sets, some structural properties of an admissible domain are investigated and also a method based on graphs is formulated that solves a combinatorial problem.…”

“…This stipulated the publication of many works devoted to the establishment of the Hamiltonianness of polyhedral graphs. It should be noted that properties of graphs and polyhedra are widely used in investigating many classes of combinatorial models and in developing new methods for solving them [1][2][3][4][5][6][7][8][9][10]. Most interesting results are obtained for polyhedra of the packing problem, maximal graph matching problem, travelling salesman problem, knapsack problem, etc.…”

“…In this article, based on the established interrelation between problems over combinatorial sets and graphs of polyhedra of corresponding sets, some structural properties of an admissible domain are investigated and also a method based on graphs is formulated that solves a combinatorial problem. In particular, in [4], a method of construction of a Hamiltonian path inside hyperfaces is described. In this article, the problem of construction of a Hamiltonian path between hyperfaces is stated, i.e., a method of construction of a Hamiltonian path is proposed that consists of "pasting" the third layer in two layers if the Hamiltonian path inside the two layers is known.…”