The W-Hierarchy

Abstract

We introduce the W-hierarchy of parameterized complexity classes in the tower

$$\mathrm{FPT}=W[0] \subseteq W[1] \subseteq W[2] \subseteq\cdots \subseteq W[t]\subseteq \cdots\subseteq W[\mathrm{SAT}]\subseteq \cdots\subseteq W[\mathrm{P}]. $$

Presumably, all the inclusions are proper. The W-Hierarchy gives us means of comparing parameterized problems that are presumably not in FPT, with respect to the “strength” of their parameterized intractability. The key idea is to use a form of circuit depth in defining the classes. We discuss the remarkable fact that “almost all” (or at least a very large percentage) of the natural parameterized problems investigated to date are precisely complete for one of the first three classes in this hierarchy, a remarkable empirical finding that supports the definitional framework. We also introduce a natural generalization called the W ^∗[t]-hierarchy, about which only a little (but for exact parameterized complexity analysis, a useful little), is so far known.