Критерії подібності динамічних задач комбінаторної оптимізації

Abstract

In combinatorial optimization, you can cite many examples when the problems from different classes are solved according to the same computational scheme. This is due to the fact that the specified problems have similarities, due to which they are solved by one method or modification of the same algorithm. It differs from geometric and described in the theory of similarity. For its establishment, an analysis of the problems of different classes is conducted in order to identify common features (criteria) that determine their similarity. Using this property allows you to develop the same methods and algorithms for their solution. The problems of combinatorial optimization, as a rule, are similar of the argument of the objective function, and the problems of combinatorics are similar on the creating and arranging of combinatorial configurations. Due to this property, their sets are generated by the same algorithm or it modification.The article describes the criterias by which the similarity of dynamic problems relating to different classes is established. The problems of combinatorial optimization, in which in the process of their solution, the current information by which the result is evaluated is generated, and the search for an optimal solution is carried out in stages with the calculation of partial amounts of the objective function, is called dynamic. The main criterias of similarity to them is the change in the result of the solution in time and for its current reference, the need to calculate the partial objective function.The process of their solution is described by a directed acyclic graph, and the partial values of the objective function change over time and are calculated according to the recurrent rules. When finding their optimal values, the Bellman principle is followed. The revealed properties of similarity, which are characteristic of the problems of this class, determine their universality, through which they are solved by the same method. Typically, dynamic programming is used to solve these problems. The study and use of this property in combinatorial optimization in the future will allow solving insoluble problems for solvable ones. Examples of some dynamic combinatorial optimization problems are given