On the Problem of a Linear Function Localization on Permutations

Abstract

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.