Fundamentals of Parameterized Complexity Revisited

Flum and Grohe define a parameter (parameterization) as a function $\kappa$ which maps words over a given alphabet to natural numbers. They require such functions to be polynomial-time computable. We show how this technical restriction can be lifted without breaking the theory. More specifically, instead of $\kappa$ we consider the set of languages that it bounds as parameterization and define the basic notions of parameterized complexity in terms of promise problems, which completely replace slices. One advantage of this formalization is that it becomes possible to interpret any complexity-theoretic concept which can be considered on a restricted set of inputs as a parameterized concept. Moreover, this formalization provides a unified way to apply the parameterization paradigm to other kinds of complexity such as enumeration or approximation by simply defining promise problems.