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. s 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.