What's the right algo might not even be a clear answer, but even just identifying something as needing a tree-like structure (rather than say a linear brute force search) could be a huge leap in the right direction. Being able to do rudimentary complexity calculations in your head helps.

Broad knowledge could even identify several different approaches to the same problem, which can be compared on strengths and weaknesses even before trying one.

E.g for a nonlinear an optimization problem you might consider simulated annealing, genetic programming, gradient descent etc. and you need to know which is easy/hard to write, that gradient is good if you have (monotone) gradients etc.