On the Classical Version of the Branch and Bound Method

In the computer literature, a lot of problems are described that can be called discrete optimization problems: from encrypting information on the Internet (including creating programs for digital cryptocurrencies) before searching for “interests” groups in social networks. Often, these problems are very difficult to solve on a computer, hence they are called “intractable”. More precisely, the possible approaches to quickly solving these problems are difficult to solve (to describe algorithms, to program); the brute force solution, as a rule, is programmed simply, but the corresponding program works much slower. Almost every one of these intractable problems can be called a mathematical model. At the same time, both the model itself and the algorithms designed to solve it are often created for one subject area, but they can also be used in many other areas. An example of such a model is the traveling salesman problem. The peculiarity of the problem is that, given the relative simplicity of its formulation, finding the optimal solution (the optimal route). This problem is very difficult and belongs to the so-called class of NP-complete problems. Moreover, according to the existing classification, the traveling salesman problem is an example of an optimization problem that is an example of the most complex subclass of this class. However, the main subject of the paper is not the problem, but the method of its soluti- on, i.e. the branch and bound method. It consists of several related heuristics, and in the monographs, such a multi-heuristic branch and bound method was apparently not previously noted: the developers of algorithms and programs should have understood this themselves. At the same time, the method itself can be applied (with minor changes) to many other discrete optimization problems. So, the classical version of branch and bound method is the main subject of this paper, but also important is the second subject, i.e. the traveling salesman problem, also in the classical formulation. The paper deals with the application of the branch and bound method in solving the traveling salesman problem, and about this application, we can also use the word “classical”. However, in addition to the classic version of this implementation, we consider some new heuristics, related to the need to develop real-time algorithms.