I was trying to find a way to solve the problem in the famous game Flow. http://moh97.us/flow/
After googling I find out that this is a NP-complete problem. A good solution would make use of heuristics and cuts. How can I spot a NP-complete problem easily? Sometimes when I block, I can't see the obvious solution. When this happens with an NP-complete, it's better to recognise it quickly and move on to the next problem.
When you have an explosion of objects (say objects whose count grows exponentially based on some parameter or parameters), this should point you in the direction that it's an NP-complete problem. When you have to inspect, check too many objects (combinatorial or others). Usually these objects are subsets or sub-spaces of some initial object space. You should build some intuition for this. But as usual, the intuition lies sometimes (I've been lied like this by my intuition on 2-3 occasions).
Then once you suspect some problem is NP-complete, just Google for it and try finding more information about the same or about a similar problem.
This is what I do at least and I've been solving quite a few algorithmic problems some time ago.
Here is a nice problem which I am pretty sure is NP-complete but which can be solved through a genetic algorithm for example.
http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=973
And as Dukeling said, there's no generic way of doing this.