Polynomial solvability of $NP$-complete problems

TL;DR

This paper demonstrates that certain NP-complete problems, including Hamiltonian cycle, can be solved in polynomial time by reducing them to linear programming and assignment problems, challenging the common belief about their computational complexity.

Contribution

It introduces a polynomial-time solution approach for NP-complete problems through reduction to linear programming and assignment problems, proposing a new perspective on their complexity.

Findings

Polynomial algorithms for problems P and L are established.

Optimal solutions for NP-complete problems are obtainable in polynomial time.

The approach challenges the traditional view of NP-complete problems' intractability.

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.