On Parameterized Exponential Time Complexity

In this paper we study the notion of parameterized exponential time complexity. We show that a parameterized problem can be solved in parameterized 2o(f (k))p(n) time if and only if it is solvable in time O(2δf (k)q(n)) for any constant δ > 0, where p and q are polynomials. We then illustrate how this equivalence can be used to show that special instances of parameterized NP-hard problems are as difficult as the general instances. For example, we show that the Planar Dominating Set problem on degree-3 graphs can be solved in 2o( √ k)p(n) parameterized time if and only if the general Planar Dominating Set problem can. Apart from their complexity theoretic implications, our results have some interesting algorithmic implications as well. © 2009 Elsevier B.V. All rights reserved.