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## Probabilistic Prediction of the Complexity of Traveling Salesman Problems Based on Approximating the Complexity Distribution from Experimental Data

We show the results of a statistical study of the complexity of the asymmetric traveling salesman problem (ATSP) obtained by processing a specially generated pool of matrices. We show that the normal distribution can serve as an approximation to the distribution of the logarithm of complexity for a fixed problem dimension. We construct a family of probability distributions that represent satisfactory approximations of the complexity distribution with a dimension of the cost matrix from 20 to 49. Our main objective is to make probabilistic predictions of the complexity of individual problems for larger values of the dimension of the cost matrix. We propose a representation of the complexity distribution that makes it possible to predict the complexity. We formulate the unification hypothesis and show directions for further study, in particular proposals on the task of clustering “complex” and “simple” ATSP problems and proposals on the task of directly predicting the complexity of a specific problem instance based on the initial cost matrix.

This book presents open optimization problems in graph theory and networks. Each chapter reflects developments in theory and applications based on Gregory Gutin’s fundamental contributions to advanced methods and techniques in combinatorial optimization.

Researchers, students, and engineers in computer science, big data, applied mathematics, operations research, algorithm design, artificial intelligence, software engineering, data analysis, industrial and systems engineering will benefit from the state-of-the-art results presented in modern graph theory and its applications to the design of efficient algorithms for optimization problems.

Topics covered in this work include:

· Algorithmic aspects of problems with disjoint cycles in graphs

· Graphs where maximal cliques and stable sets intersect

· The maximum independent set problem with special classes

· A general technique for heuristic algorithms for optimization problems

· The network design problem with cut constraints

· Algorithms for computing the frustration index of a signed graph

· A heuristic approach for studying the patrol problem on a graph

· Minimum possible sum and product of the proper connection number

· Structural and algorithmic results on branchings in digraphs

· Improved upper bounds for Korkel--Ghosh benchmark SPLP instances

It is shown that the logarithm of the complexity (number of nodes in the decision tree of a branch and bound algorithm) of the individual traveling salesman problem is approximately normally distributed. We use a linear regression model (logarithm of the complexity — standard normal distribution) to estimate parameters of normal distribution, which fit the sample. Borders of the interval, which contains 90% of the sample of the logarithm of the complexity, are also given.

In the paper we consider the hidden parameter (measurement time t_meas) which combines quantum and classical theory. We show that the Bose–Einstein and Fermi–Dirac quantum distributions turn out to be the decisive factor in the construction of isotherms in classical thermodynamics and in the description of the phase transition “gas to liquid” and “liquid to solid”.

For practical, important tasks in the fi elds of economics and logistics, as well as in a number of technical applications, it becomes necessary to solve the traveling salesman problem (TSP). Quite often, the features of these problems lead to the traveling salesman problem in asymmetric formulation (asymmetric traveling salesman problem, ATSP). Moreover, in some practical applications it is desirable to obtain an exact solution. One of the known exact algorithms for solving the ATSP is an algorithm that implements the well-known branch and bound method. The known experimental estimates of its complexity on the average are exponential. However, this does not mean that for small dimensions of the problem (currently, no more than 70–75), the expected time for solving the individual problem is unacceptably high. The need to reduce the time for solving individual problems dictated by practice is associated with the use of various modifi cations of this algorithm, of which a modifi cation that involves storing truncated matrices in the search decision tree is one of the most eff ective. In this article, the authors rely on this modifi cation. Other possible improvements in the time effi ciency of the software implementation of the branch and bound method are related, among other things, to obtaining the initial approximation by heuristic algorithms. As a result, we get a combined algorithm, in which, at the fi rst stage, some heuristics works to obtain the initial solution, from which the branch and bound method starts. This idea has been discussed for a long time, but the problem is that to reduce time, such a heuristic algorithm is needed that delivers a solution close to optimal which will be found quite fast. One of the possible solutions to this problem is the subject of this article. The subject of the research in this article is the choice of the best heuristic algorithm which, when applied, leads to an increase in temporal effi ciency in combination with the algorithm of the branch and bound method, and an experimental study of its software implementation in order to obtain an average time for solving individual problems. On the basis of the results obtained, recommendations are given on the limiting dimensions of the problem that allow for an acceptable solution time, something which is of interest in the practical application of this combined algorithm in the tasks of business informatics and logistics.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

Existing approaches suggest that IT strategy should be a reflection of business strategy. However, actually organisations do not often follow business strategy even if it is formally declared. In these conditions, IT strategy can be viewed not as a plan, but as an organisational shared view on the role of information systems. This approach generally reflects only a top-down perspective of IT strategy. So, it can be supplemented by a strategic behaviour pattern (i.e., more or less standard response to a changes that is formed as result of previous experience) to implement bottom-up approach. Two components that can help to establish effective reaction regarding new initiatives in IT are proposed here: model of IT-related decision making, and efficiency measurement metric to estimate maturity of business processes and appropriate IT. Usage of proposed tools is demonstrated in practical cases.