I have a P/NP question.
I have read that were any NP problem be found to have a polynomial time algorithm, then we can reduce any other NP problem to a form where we can use our first algorithm to solve the new NP problem in polynomial time as well.
I don't see why this is the case.
Example: Hamiltonian Cycle problem, and Traveling salesman.
Were I to find a polynomial time algorithm to determine the existence of a hamiltonian cycle in a graph, I still don't see how this could let me find a TSP tour in an upper bound of polynomial time. The amount of subgraphs we would have to iteratively check for a hamiltonian cycle is often exponential. It's typically impossible to create a single subgraph that contains all potential TSP tours under a given bound and would still only contain one hamiltonian circuit.