All NP-complete problems can be transformed into each other using a polynomial time reduction. Therefore if you have a polynomial time solution to one, you imm

All NP-complete problems can be transformed into each other using a polynomial time reduction. Therefore if you have a polynomial time solution to one, you immediately have a polynomial time solution to all of them. A hypothetical straightforward way to show P = NP would be to provide a polynomial time solution to one of the NP-complete problems, which, again, would therefore immediately solve them all in polynomial time. However, it is possible to prove that a method of solving a problem exists without exhibiting it. For example, some algorithms depend on expander graphs existing (such as multi-butterflies); initially expander graphs could be proven to exist (using Hungarian probabilistic techniques), but such a proof does not tell you an practical method for making an expander graph.