ON THE THEORY OF NP-COMPLETE PROBLEMS

It is shown that the theory of NP -сomplete problems can't be used for the analysis of optimizing tasks.


INTRODUCTION
Now seven mathematical problems of the tasks of the millennium included the list one of them are known it is a task about interrelation of classes P and NP. The question of interrelation of classes P and NP, is considered now one of the main open questions of modern mathematics and theoretical cybernetics. Founders of this problem are Stephen Arthur Cook the professor of university from Toronto, the winner of an award of Turing and professor Leonid Levin. In Cook's works the concept of of NP -сomplete problems was entered and is proved that the task "feasibility" known still as a SAT task is a universal NP -сomplete task. Further development of the theory of NP -сomplete tasks was carried out by the professor of the Harward university Richard Meningom Karp. Incentive motive for writing of this short message, were works of the Indian mathematician Vineya Deolalikar and the article "On the Relationship between Classes P and NP" of the Ukrainian professor of Anatoly D. Plotnikov in Journal of Computer Science 8 (7): 1036-1040, 2012. In these works it is proved, the fact of discrepancy of these classes. The purpose of this short message is to show an incorrectness of these attempts of the proof on the basis of results already received by the author in works [4,5] and results received in Lavrov and Zykova's works [6,7,8].

PROBLEM SETTING AND SOLUTION
The theory of NP-complete problems is constructed for problems of recognition of properties. The problem of recognition L can be considered as consisting of two sets: Di and Yд, where Diset of all single problems, and Yдset of problems with the answer "yes", thus Yд Di. The form of these problems consists of two parts. In the first part the exposition of conditions of the problem in terms of various components is given: sets, networks, numbers etc. In the second part the question assuming one of two answers "yes" or "no" is formulated. Informally, class of NP-complete problems is defined by means of concept of nondeterministic algorithm. Such algorithm consists of two stages: guessing and check. At first, under the set single problem I a guessing of structure S takes place, and further, taking into account statements of problem I, check by the determined algorithm which is ended either by the answer "yes" or "no" is carried out. As it is shown in [1], nondeterministic algorithm solves the problem of recognition L, if for any single problem I Di two following conditions are met: 1. If I Yд there is such structure S which guessing leads to that the check stage will be completed by the answer "yes".
2. If I Yд there is no such structure S which guessing for I will lead to that the check stage will be completed by the answer «no».
The concept of polynomial "checkability" [1] allows actual selection of a class of NP-complete problems, and in addition, "checkability" for polynomial time does not attract decidability of the problem of recognition for polynomial time.
The problem of recognition L is called NP -complete if L NP and any other problem L / from this class is reduced to L polynomially.
Let's assume that the polynomial algorithm for solution of some NP-complete class problem I is obtained, then according to Cook theorem [1,2,3] it follows polynomial resolvability of all NP-complete class problems as it appears from Cook theorem they are polynomially reduced to each other. However in the study [4] it is shown that Cook theorem is incorrect also a class of NP-complete problems is introduced incorrectly. Therefore, polynomial resolvability of problems called in studies of Cook and Karp [1,2,3] as NP-complete does not follow from polynomial resolvability of the above mentioned problem as far as in the studies [4,5] it is shown that all problems which related to NP-complete class can be divided into subsets of problems i u within which polynomial reducibility is possible and the hypothesis is stated that polynomial reductions between subsets i u of separate individual problems only are most likely possible. Therefore it is possible to speak here only about polynomial resolvability of individual problems which can be reduced to the problem I. As it follows from [1,6,7,8] the fact supports the given hypothesis that now the list of NP-complete problems includes more than three thousand problems, and practically all main problems of graph theory inclusively. Then, proceeding from the polynomial reducibility of problems within this class declared by Cook, for solution of all problems of graph theory listed there should be one algorithm for their deciding with some arbitrarily high complexity that is stipulated by their polynomial reducibility to each other, but this contradicts the outcomes obtained in studies of I. A. Lavrov (1963) [8] and A.A. Zykov (1969) [6] where impossibility of construction of such algorithm is shown.
As appears from [4,5] set of objects which are described by impracticable Boolean functions in exponential number of times surpass number of objects which are described by feasible Boolean functions, and properties of polynomial reducibility are by default transferred and to the objects described by impracticable Boolean functions. Thus, Cook's theorem is fair only for objects described by feasible Boolean functions. Thus it must be kept in mind that the number of such objects isn't enough. Now some words about the proof of NP completeness of any task. As shown in studies [1,2,3], polynomial reducibility of the problem of recognition I1 to the problem of recognition I2 means availability of function f which transforms a subset of problems Di1 into a subset of problems Di2 (Di1 Di2), on the basis of some rule Пi and, thus, satisfies to two conditions:

CONCLUSIONS
The problem about correlation of classes Р and NP set by Cook and entered into the Millennium problems list is the simply incorrectly set mathematical problem therefore it is no wonder that it was possible to nobody to solve it. Therefore the problem should be eliminated from the Millennium problems list as scientists spend precious time for its solution to this day, studies of Indian mathematician Viney Deolalikar and paper "On the Relationship between P and NP Classes" of Ukrainian professor Anatoly D. Plotnikov in Journal of Computer Science 8 (7: 1036-1040), 2012 testify to it, i.e. presence of the given problem at the Millennium problems list thwarts further progress of mathematics. It is necessary to note also, that all outcomes in the theory of algorithms based on the "total" reducibility in NP-complete class problems declared by Cook should be revised seriously.