  ${ NP}$-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.
