Say I want to solve a set of problems. I know that I can map my problem as a problem that is known to be hard in the general case (say, inference in Bayesian networks). But my set only contains instances which will map to easy instances - (say, having a tree-structure when cast to a BN inference problem)
I know BNs with tree-structures have linear-time inference algorithms. If I didn't, I may have given up on exact solutions in favour of loose approximations.
(2nd, looser example: I find myself needing to solve SAT, but turns out I can easily reduce all my problem instances to 2-SAT)
Is there a common approach to determine whether a given subset of a hard problem is hard or not? How does one typically approach this? Are these well studied