# But why *is* $FPT$ a subset of $W[1]$?

$$FPT$$ is the set of parameterized problems that are fixed-parameter tractable. If $$L_{w,h}$$ is the language associated to boolean circuits of weft $$w$$ and depth $$h$$, then $$W[t]$$ is the set of parameterized problems that can be fixed-parameter reduced to $$L_{t,h}$$ for some constant $$h$$. (I know that there are multiple equivalent definitions.) It is folklore that

$$FPT \subseteq W[1] \subseteq W[2] \subseteq \dots$$

but I do not really understand the first inclusion, and I could not find a satisfying explanation.

Niedermeier writes in "Invitation to Fixed-Parameter Algorithms" that "The inclusion $$FPT \subseteq W[1]$$ directly follows from the fact that $$W[1]$$ allows '$$FPT$$-computations' through the use of parameterized reductions". What is meant by that?

A problem $$P$$ in FPT has a parametrized reduction to any parametrized decision problem that has at least one "true" and "false" instance. The reduction algorithm has enough time to compute whether an instance of $$P$$ is true or false (because $$P$$ is in FPT), and can then simply map it to one of the true or false instances of the other language.
So, the problem $$P$$ has a parametrized reduction to $$L_{1,1}$$ in particular, and therefore is in $$W[1]$$.