Bounded fixed-parameter tractability and reducibility

We study a refined framwork of parameterized complexity theory where the parameter dependendence 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.

• Type: Journal
• Year: 2007
• Where: APAL
• Authors: Rod Downey, Jörg Flum, Martin Grohe, Mark Weyer
• Keywords: APAL 2007, F-fixed Parameter Tractability, Parameter Tractability, Parameterized Complexity Theory