Can you solve arbitrary traveling salesman problem with that? There is only a polynomial number of required laser cuts (n nodes, n * (n-1) edges) and the source can be next to sink with a fake 0 edge between them.
I say "approximate" because, as has been discussed in this thread, the current actually follows every possible path to some extent, weighted by their resistance. So when there multiple branching paths with similar costs, they will have similar current flow, and the current along each path is not constant. So in general it can be difficult to find the exact minimum resistance path just by measuring the current.
It's not going to save you any time, since wiring up every path is basically the same as brute forcing every path anyway, but it would work.
Yes, you can solve TSP in linear time using electricity.
First step is to construct a circuit with a single source, and multiple independent paths -- one for each possible TSP graph traversal. Constructing this circuit can take a bit of time, but after that, finding the shortest path will be extremely fast.
Essentially an analog computer. Some of the reason people want quantum or DNA computers to exploit physical mechanisms beyond stacks of binary gates.