Observations and Applications of Element Transfers in Sets

Key words: NP-class problems, discrete optimization problems, approximate solution methods, traveling agent problem, processor, program execution scheduling, mass distribution, branch and bound algorithm No more efficient algorithms than complete search algorithms are known for solving many discrete optimization problems. Such problems include NP-class problems. In order to solve these problems, it is often necessary to search for the optimal solution among all transfers of the elements of a set, among all subsets of a set and all possible partitions of natural numbers, and so on. In such cases, to solve the problem, it is essential to utilize algorithms that observe all transfers of elements of the set, all subsets of the set, all possible partitions of a natural number, and similar algorithms. The main topic of this work is the systematic observation of transfers of all elements of a set. Each algorithm for systematic observation of transfers must include the selection of an initial transfer, the transition from one transfer to the next one, and the condition for the algorithm’s completion. The following work develops an algorithm for the observation of transfers of all elements of a set. The paper considers the problems of mass distribution, distribution of program execution on one processor, and the traveling salesman problem, which are solved by considering all transfers of elements of the set. The algorithm developed in the paper can be used to solve the above-mentioned and equivalent problems.