The most interesting thing to come out of the backtracking algorithm is the fact
that it is empirically close to 
, this shows that even looking at less
than 1% of the tree or no matter what we can remove from the tree the time to
search it is going to take exponential time. I think that to see any kind of
real improvement in the time taken to solve this problem, that makes it actually
a useful tool, an algorithm that doesn't use a tree structure needs to be used.
Innovation comes from looking at a problem in a different way where you can easily see the solution is simpler. Such as when I changed my view of this problem from a list of numbers that I needed count through to a tree of partial schedules I immediately saw that a large portion of that tree was not necessary and large speedup would occur. To continue finding a better algorithm we need to find a better way of looking at the data where a solution just drops out.