Polynomial solvability of $NP$-complete problems

Abstract: ${ NP}$-complete problem "Hamiltonian cycle"\ for graph $G=(V,E)$ is extended to the "Hamiltonian Complement of the Graph"\ problem of finding the minimal cardinality set $H$ containing additional edges so that graph $G=(V,E\cup H)$ is Hamiltonian. The solving of "Hamiltonian Complement of a Graph"\ problem is reduced to the linear programming problem {\bf P}, which has an optimal integer solution. The optimal integer solution of {\bf P} is found for any its optimal solution by solving the linear assignment problem {\bf L}. The existence of polynomial algorithms for problems {\bf P} and {\bf L} proves the polynomial solvability of ${ NP}$-complete problems.