$FPT$ 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?


1 Answer 1


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 $W[1]$.


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