The resource efficiency of different implementations of the branch-and-bound method for the classical traveling salesman problem depends, inter alia, on ways to organize a search decision tree generated by this method. The classic «time-memory» dilemma is realized herein either by an option of storing reduced matrices at the points of the decision tree, which leads to reduction in the complexity with additional capacity cost, or matrix recalculation for the current node, which leads to an increase in complexity while saving memory. The subject of this paper is an experimental study of temporal characteristics of solving the traveling salesman problem by the branch-and-bound method to identify a real reduction of span time using additional memory in a selected structure of a decision tree. The ultimate objective of the research is to formulate recommendations for implementing the method in practical problems encountered in logistics and business informatics. On the basis of experimental data, this paper shows that both considered options of the classic algorithm for the traveling salesman problem by the branch-and-bound method generate software implementations with an exponential dependence on the execution time of the input length. The experimental results permit us to suggest that the applicability of an additional memory capacity of no more than 1 GB results in a significant (up to five times) reduction of the time span. The estimate of the resulting trend makes it possible to recommend practical application of the software implementation of the branch-and-bound algorithm with storage of matrices-with a really available 16 GB random-access memory and with limitation of the expected average computation time of about one minute on modern personal computers whereby problems having a dimension no more than 70 can be solved exactly.

Forecasting of time characteristics of efficient implementations of the branch and bound method for the traveling salesman problem relying on characteristics of random matrixes and identification of generated times distribution") MATHEMATICAL METHODS AND ALGORITHMS OF BUSINESS INFORMATICS

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The exact algorithm that implements the Branch and Boimd method with precomputed tour which is calculated by Lin-Kernighan-Helsgaun metaheuristic algorithm for solving the Traveling Salesman Problem is concerned here. Reducing the number of decision tree nodes, which are created by the Branches and Bound method, due to a "good" precomputed tour leads to the classical balancing dilemma of time costs. A tour that is close to optimal one takes time, even when the Lin-Kernighan-Helsgaun algorithm is used, however it reduces the working time of the Branch and Bound method. The problem of determining the scope of such a combined algorithm arises. In this article it is solved by using a special characteristic of the individual Traveling Salesman Problem — the number of changes tracing direction in the search decision tree generated by the Branch and Bound Method. The use of this characteristic allowed to divide individual tasks into three categories, for which, based on experimental data, recommendations of the combined algorithm usage are formulated. Based on the data obtained in a computational experiment (in range from 30 to 45), it is recommended to use a combined algorithm for category III problems starting with n = 36, and for category II problems starting with n = 42.

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