About one class polynomial problems with not polynomial certificates

About one class polynomial problems with not polynomial certificates Kochkarev B.S. Abstract. We build a class of polynomial problems with not polynomial certificates. The parameter concerning which are defined efficiency of corresponding algorithms is the number n of elements of the set has used at construction of combinatory objects (families of subsets) with necessary properties. Let Σ = {0, 1} is set of two elements. An language L over Σ is any set of strings made up of symbols from Σ. We denote the empty string by e, and the empty language by ∅. The language of all strings over Σ is denoted Σ *. Every language L over Σ is a subset of Σ *. Let n is a natural number and S is the ordered set (a 1 , a 2 ,. .. , a n). In the capacity of combinatorial objects we will to consider the families of subsets of the set S with necessary properties. Any subset A of S can be present in look of string from n elements (σ 1 , σ 2 ,. .. , σ n), where σ i = 1, if a i ∈ A and σ i = 0, if a i / ∈ A. In the capacity of languages we will consider the families of subsets of the set S with definite properties. Thus a language in our case is a subset of the strings (σ have entered a concept of complexity class P. Definition 1 [1,2]. A language L belong to P if there is an algorithm A that decide L in polynomial time (≤ O(n k)) for a constant k. Class of problems P is called polynomial. According to [3] J. Edmonds has entered also the complexity class N P. This is the class of problems (languages) that can be verified by a polynomial-time algorithm. Definition 2 [3]. A language L belongs to N P if there exists a two-input polynomial-time algorithm A and such polynomial p(x) with whole coefficients that L = {x ∈ {0, 1} n : there exists a certif icate y with | y |≤ p(| x |) and A(x, y) = 1} In this case we say that the algorithm A verifies language L in polynomial time. According to definition 2 if L ∈ P and | y |≤ p(| x |) then L ∈ N P. But if L ∈ P …