> 2. When there's some large set of instances of some NP-hard problem that are tractably solvable in practice (like SAT), the importance of that is that there's some non-NP-hard subset here. Indeed, SAT is FPT (fixed parameter tractable [1]), an "easier" type of NP, for which decomposition can help. In contrast, graph colouring is thought to not be FPT.
Sorry but I need to clarify here. "SAT is FPT" does not mean anything. FPT only makes sense when you tell what is the *parameter*. Every problem is FPT when parametrized by the input size so graph colouring and SAT are FPT wrt to the size of the input (the graph and the formula respectively). What you meant: graph colouring parametrized by the number of colours is unlikely to be FPT (since it is W[1]-hard). SAT is FPT for many parameters such as treewidth (of the formula). Oh, and btw, graph colouring is also FPT when parametrized by treewidth (of the graph).
> 2. When there's some large set of instances of some NP-hard problem that are tractably solvable in practice (like SAT), the importance of that is that there's some non-NP-hard subset here. Indeed, SAT is FPT (fixed parameter tractable [1]), an "easier" type of NP, for which decomposition can help. In contrast, graph colouring is thought to not be FPT.
Sorry but I need to clarify here. "SAT is FPT" does not mean anything. FPT only makes sense when you tell what is the *parameter*. Every problem is FPT when parametrized by the input size so graph colouring and SAT are FPT wrt to the size of the input (the graph and the formula respectively). What you meant: graph colouring parametrized by the number of colours is unlikely to be FPT (since it is W[1]-hard). SAT is FPT for many parameters such as treewidth (of the formula). Oh, and btw, graph colouring is also FPT when parametrized by treewidth (of the graph).