Cybernetics and Computer Engineering, 2022, 1(207)
Gritsenko V.I., Corresponding Member of NAS of Ukraine,
Honorary Director of International Research and Training Center for Information Technologies and Systems of the National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine
Tymofijeva N.K., DSc (Engineering), Senior Researcher,
Acting Head of Department of Complex Research of Information Technologies
International Research and Training Center for
Information Technologies and Systems of the
National Academy of Sciences of Ukraine
and Ministry of Education and Science of Ukraine,
40, Acad. Glushkov av., Kyiv, 03187, Ukraine
FINDING SUBCLASSES OF SOLVABLE PROBLEMS IN COMBINAR OPTIMIZATION AND ARTIFICIAL INTELLIGENCE BY STRUCTURE OF INPUT INFORMATION
Introduction. The literature for some classes of combinatorial optimization problems describes subclasses that have a certain structure of input data with a clear nature, for which there is a known method of analytical finding of a global solution without searching for options. These subclasses of problems are called solvable. They can be used to reduce unsolvable combinatorial optimization problems to solvable ones.
The purpose of the paper is to identify the following main approaches for solving combinatorial optimization problems: a) methods and algorithms based on partial search of variants; b) methods and algorithms based on recognizing the structure of input information. The second approach includes work on finding subclasses of solvable problems and development of recognition algorithms according to these subclasses of the structure of input information. The problem is to identify subclasses for different classes of combinatorial optimization problems according to the structure of input data, for which according to the developed rules analytically find a global solution.
Methods. To select subclasses of solvable problems, we use the method of modeling input data by functions of a natural argument. To do this, the input data, which are given by finite sequences, are given by the functions of the natural argument, one of which is combinatorial. For various such functions, which are represented by linear, periodic, convex, the global values of the objective function are determined, both maximum and minimum.
Results. Subclasses of solvable problems are distinguished for different classes of combinatorial optimization and artificial intelligence problems according to the structure of input data. Found global maximum and minimum for assignment problems, traveling salesman problem, placement of objects on a given surface.
Conclusions. Using the method of modeling the structure of input data by means of natural argument functions allows to reduce some unsolvable problems of combinatorial optimization to solvable ones. For the latter, it is easy to find an argument (combinatorial configuration) for which the objective function acquires a global minimum and maximum, as well as to formulate the expression behind which is its value. In artificial intelligence problems, the subclasses of solvable problems are distinguished both on the basis of similarity and the structure of the input data. Using them allows to reduce unsolvable problems to solvable ones.
Keywords: subclasses of solvable problems, natural argument function, combinatorial optimization, similarity measures, objective function.
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