I'm currently studying the travelling salesman problem and Christofides algorithm. I think I understand that TSP is an NP-hard problem, and so the complexity of calculating a solution grows exponentially as we add more destinations to the problem.
I'm not well versed in computer science, or computational complexity theory, but upon reading the 'NP-hardness' wikipedia page it says that
"It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven."
What I don't get is that Christofides algorithm is considered a polynomial time approximation algorithm, which finds approximations to the optimal solution of the travelling salesman problem. Doesn't this go against the quotation above (i.e. isn't this an example of a polynomial time algorithm being used to solve an NP-hard problem?). Hope this is clear.