In the following paper :
Czopik, J. (2019) An Application of the Hungarian Algorithm to Solve Traveling Salesman Problem. American Journal of Computational Mathematics,9, 61-67.
In the Introduction, the author claims that the problem is NP-hard. They proceed to give a exact solution in polynomial time.
My problem is that if we can solve TSP optimization in polynomial time, we can also solve the decision version of TSP, which is NP-complete in polynomial time.
The TSP decision problem asks that given the costs and a number $x$, decide whether there is a round-trip route cheaper than $x$. let us denote $min_x$ as the smallest path that satisfies our conditions. we can find min_x from the above claim in polynomial time. Then if $x > min_x$, the answer is Yes else No. Thus, the decision problem can be solved in polynomial time.
But TSP-decision belongs to NP-complete and thus there cannot be a polynomial time solution. Where am I getting things wrong?