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 familyF . 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 ofF 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.