Bounded Fixed-Parameter Tractability and Limited Nondeterminism

Abstract

In the definition of fixed-parameter tractability we allowed arbitrary computable functions $f$ to bound the dependence on the parameter of the running time $f(\kappa(x))\cdot |x|^{O(1)}$ of an fpt-algorithm. This liberal definition was mainly justified by the hope that "natural" problems in FPT will have "low" parameter dependence. While this is true for many problems, in Theorem 10.18 we saw that there is no elementary function bounding the parameter dependence of $p$-MC(TREE,FO), the parameterized model-checking of first-order logic on the class of trees. There are viable alternatives to the notion of fixed-parameter tractability obtained by simply putting upper bounds on the growth of the parameter dependence $f$. Natural choices are $f\in 2^{O(k)}$ and $f\in 2^{k^{O(1)}}$.