Is there any NP-hard problem that we can find a mechanical "polynomial time" solution to? For example, suppose we construct a graph out of something physical, e.g. we have have pipes through which we can move water. If this pipe system was suitably built, could we solve say some routing problem by seeing if water flows from one node to another?
So by a mechanical solution, I mean a physical configuration and not a computer program, i.e. I am not looking for a computer algorithm. I also used "polynomial mechanical solution" to describe a solution on a physical device that runs in time polynomial in the order of the input. For example, a piping system could yield the answer in "number of nodes squared" seconds.