Bounded fixed-parameter tractability and reducibility

We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family ℱ. For every family ℱ of functions, this yields a notion of ℱ-fixed-parameter tractability. If ℱ is the class of all polynomially bounded functions, then ℱ-fixed-parameter tractability coincides with polynomial time decidability and if ℱ is the class of all computable functions, ℱ-fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices of ℱ between these two extremes, for example the class of all singly exponential functions.In this article, we study the general theory of ℱ-fixed-parameter tractability. We introduce a generic notion of reduction and two classes ℱ-W[P] and ℱ-XP, which may be viewed as analogues of NP and EXPTIME, respectively, in the world of ℱ-fixed-parameter tractability.