Ivan V. Sergienko

Finding maximum independent sets in graphs arising from coding theory

New results are presented concerning binary correcting codes, such as deletion-correcting codes, transposition-correction codes, and codes for the Z-channel. These codes are important due to the possibility of packet loss and corruption on internet transmissions. It is known that the problem of finding the largest correcting codes can be reduced to a well-known combinatorial optimization problem on graphs, the maximum independent set problem. A general scheme of organizing a local search for the maximum independent set problem is discussed. Based on the developed heuristics, an exact branch-and-bound algorithm is proposed, which is able to find exact solutions for graphs with over 500 vertices within a reasonable time.

Estimating the size of correcting codes using extremal graph problems

Some of the fundamental problems in coding theory can be formulated as extremal graph problems. Finding estimates of the size of correcting codes is important from both theoretical and practical perspectives. We solve the problem of finding the largest correcting codes using previously developed algorithms for optimization problems in graphs. We report new exact solutions and estimates.

Combinatorial Optimization Problems

We consider combinatorial optimization problems (COP), i.e., finding extrema of an objective function on a combinatorial space. Many various important applied and theoretical problems of different degree of complexity can be presented as problems in graph theory. These, for example, are the problem of designing communication networks, engineering devices, electronic and digital equipment, traffic control, planning and management of industrial and commercial activities, layout of equipment. To solve them, the number graph theory is proposed. To solve other COP, the theories of mathematical safes, construction of discrete images, combinatorial recognition, and solution of extremum problems on combinatorial configurations are proposed. The theory of mathematical safes analyzes a number of problems on special graphs, which are called safes. One should use certain operations to decode the states of vertices called locks. It is used in coding theory, information security, and cryptology. The theory of constructing discrete images involves problems where it is necessary to use a given set of colored patterns to construct a prescribed image on a rectangular field of view. It is also used in coding theory and information security. The combinatorial recognition theory addresses problems of recognition of the properties of individual objects from some set by so-called tests or trials, which allow a wide interpretation. Such problems occur in various lotteries, banking sector, and insurance companies. The graph-theoretical approach is proposed to solve these problems. Many problems in economic and mathematical modeling reduce to optimization problems on combinatorial configurations such as permutations, partitions, combinations, arrangements, multisets, etc. To solve them, the direct structuring method based on the graph theory is proposed.

Mathematical Modeling and Information Technologies for the Analysis of Economic Processes

We consider information technologies that include the following interindustry aggregated balance models to identify the main lines of structural and technological changes: economic dynamics given imperfect competition in the labor market; optimal servicing of the public debt of Ukraine; planning of regional budgets and allocation of intergovernmental transfers; and sustainable environmental, social–institutional, and economic development of the society. To carry out calculations with these models, numerical algorithms were selected, modified, and improved, and computer programs were created for SCIT supercomputers. The chapter analyzes the influence of the own resources of regions and interbudgetary transfers on funding three major social programs: education, health, and social protection. A number of hypotheses and proposals concerning the optimization of the system of intergovernmental transfers are formulated. The redistributive properties of the system of intergovernmental transfers in Ukraine are investigated.

Optimization Methods for Solving Problems of Transcomputational Complexity

This chapter addresses subgradient nonsmooth optimization methods. The generalized gradient descent method known later as the subgradient method is considered. It was used to solve many high-dimensional problems of production and transportation scheduling with decomposition schemes. Subgradient methods with space dilation, whose special cases are the ellipsoid method and r-algorithms, are analyzed. The application of nonsmooth optimization methods to various mathematical programming problems is discussed. The algorithmic scheme of the well-known descent method is considered. This scheme has made it possible to develop a wide range of local search algorithms, which were used in application packages elaborated at the Institute of Cybernetics. The modern method of local type, the probabilistic method of global equilibrium search, is analyzed. It was applied to solve various classes of problems of transcomputational complexity and appeared to be the most efficient discrete optimization technique. The approaches to accelerating the solution of discrete optimization problems are discussed. The stability of vector discrete optimization problems is analyzed. Different types of stability of integer optimization problems and regularization of unstable problems are considered. Some of the chapter deals with stochastic optimization that involves decision making under risk and uncertainty. Stochastic quasigradient methods and their modifications for stochastic programming problems are considered. The solution of stochastic global optimization, stochastic discrete optimization, stochastic minimax, and vector stochastic optimization problems is analyzed. The use of stochastic optimization methods in risk assessment and management is discussed. The chapter reviews the results of the mathematical theory of control of distributed parameter systems and stochastic and discrete systems and addresses the robust stability, determination of invariant sets of dynamic systems, and spacecraft control. The methods of the analysis of dynamic games, including those with groups of participants under various constraints, are considered as the development of motion control methods.

Modeling of Imperfect Competition on the Labor Market

The social situation of last decades in post-communist countries (especially in countries of the former Soviet Union) has been certainly an extremely difficult one. Various grave problems existed: rising inequalities in income both at downward and upward stages of the business cycle show up in islands of prosperity in the sea of distress and poverty, as well as in capital flights and growing indebtedness. All of these factors lead to the criminalization of society. Extremely low wages are the main reason of such a situation. Theoretical analysis of labor market is necessary to understand essential issues of this situation.

Information Technologies as Tools for Studying Complex Processes

The chapter refers to the development of high-performance supercomputer systems, reconfigurable knowledge-oriented computer systems, problem-oriented systems for digital signal processing, computer instrumentation, model-oriented technologies for real-time management of complex technological objects, computer technologies, medico-biological and agroecological computer devices and sensor networks, automation of border and migration control in Ukraine, and creation of situation centers. The development of modern information protection tools in computer technologies, personalization of information, and new Ukrainian mechanically changeable hardware tools to authenticate users of information systems and intelligent mobile devices are also considered. Attention is paid to the information technology of developing the human interface with computer environment in a text, voice, and visual forms.

Theoretical and Applied Programming

The chapter presents the main results of the automatic–algebraic approach proposed by Glushkov and his followers in theoretical and applied programming, which laid the foundation of the Kyiv School of Cybernetics and Informatics. Among the main results are those on automata theory, theory of discrete transforms, algebra of algorithms, and their application to developing the paradigms of structural, algebraic, insertion programming, etc. This approach is shown to be efficiently applicable to the design of electronic computing machines and systems (MIR series machines, macro-pipeline computer systems). The chapter presents results on parallel computing and parallel programming and demonstrates models, methods, and technologies of macro-pipeline parallel programming, including those for multiprocessor supercomputer systems (clusters). The development of software engineering, construction of effective operating environments and automation tools for the design of application software systems are also considered. Attention is paid to the role of K. L. Yushchenko and her school in the development of programming in Ukraine.

Computer Technologies in Medical and Biological Research

The chapter highlights interesting results obtained in decoding the genetic information for models of DNA, proteins, and important processes in cells of the human body. The symmetry rules are established, and genetic information is shown to be encoded based on the optimal Watson–Crick model. The properties of the symmetry are analyzed for amino acid sequences of proteins along two opposite strands in genomes of higher organisms. A model is developed to identify functional regions of genes in DNA based on Markov models with hidden variables. Methods for modeling of biological processes in cells of living organisms using the apparatus of active particles for which transformations of random walk, composition of a complex particle from basic particles, and disintegration of complex particles are carried out. Intelligent technologies of telemedicine with the use of electronic medical advice are developed. New interdisciplinary research areas of bioecomedicine and digital medicine are considered. Microelectronic digital medicine devices for early diagnosis of cardiovascular disease, diabetes, and rehabilitation of patients with motor function disorder are described. Supersensitive magnetometric systems for research in cardiology (magnetocardiography) and for the distribution analysis of iron compounds in biological objects are created based on superconducting quantum interference device (SQUID). A feature of these SQUID magnetic systems is their noise immunity, which allows carrying out research in unshielded premises. Magnetocardiographic systems were used to develop a new screening information technology for early diagnosis of heart disease. It was shown for the first time that a harmful liquid introduced intravenously into a laboratory animal yields iron overload in the liver. Smart sensors were developed for biomedical purposes. These are noninvasive hemoglobin meter, a device for the assessment of the peripheral blood flow, a pulse wave recording device, gas analyzers, and biosensors based on surface plasmon resonance.

Mathematical Modeling and Analysis of Complex Processes

The chapter addresses some aspects of modern computational mathematics concerning the mathematical modeling of processes in multicomponent media, wave processes in homogeneous and inhomogeneous infinite waveguides, and new methods for solving ill-posed problems of linear algebra. A methodology is proposed for the analysis of multicomponent bodies with technological or natural thin layers. These layers in many cases have a considerable effect on physical processes throughout the body. New mathematical models of processes in multicomponent media formulated as various classes of boundary-value and initial–boundary-value problems for partial differential equations with conjugation conditions are constructed and analyzed. For these problems (with discontinuous solutions), highly accurate computational algorithms are developed and efficient procedures of gradient methods for solving inverse problems are proposed and analyzed. They form the theoretical basis for modern information technologies of the analysis of complex processes. The results of investigations in the mathematical modeling of the dynamics of distributed process of the environment are described. The structure of the developed system for the automated solution of environmental problems is presented. A technique is proposed and tools are created for mathematical modeling of the distribution and formation of acoustic fields based on wave equations with complex nonself-adjoint operators. The stability analysis with respect to the initial data is carried out and the stability conditions are established for explicit and implicit three-layer difference schemes with such operators. New results in the theory of weighted pseudoinversion are obtained and used to develop efficient methods to solve ill-posed problems of linear algebra.