Recently, when I self-learnt Discrete Mathematics and Its Applications 8th by Kenneth Rosen, I had some questions about some statements in it.
Fur- thermore, if a polynomial worst-case time complexity algorithm were discovered for the traveling salesperson problem, many other difficult problems would also be solvable using polynomial worst-case time complexity algorithms (such as determining whether a proposition in n vari- ables is a tautology, discussed in Chapter 1). This follows from the theory of NP-completeness.
A practical approach to the traveling salesperson problem when there are many vertices to visit is to use an approximation algorithm. ... That is, they may produce a Hamilton circuit with total weight W′ such that $W \le W′ \le cW$, where W is the total length of an exact solution and c is a constant ... If such an algorithm existed, this would show that the class P would be the same as the class NP, perhaps the most famous open question about the complexity of algorithms
I knew that NP can only be "verifiable in polynomial time by a deterministic Turing machine" while P can be solved in polynomial time by a deterministic Turing machine.
Then if we find one approximation algorithm for NP-complete which adds the NP-hard property for NP, then it only approximates instead of solving the problem. If it can solve, then based on NP-complete, all NP problems can be solved.
I also referred to one QA answer which states:
Proving an upper bound on the possible approximation is akin to P=NP
But the answer seems to not answer the reasons behind the relation between the approximation algorithm and $P=NP$. This is weaker than the above finding both the lower bound $W$ and the upper bound $cW$.
Q:
How does the approximation algorithm for one NP-complete problem show that "we can show the existence of an algorithm that solves any given instance of problem in NP in polynomial time", i.e. $P=NP$?
