A general method to speed up fixed-parameter-tractable algorithms

A fixed-parameter-tractable algorithm, or FPT algorithm for short, gets an instance (I, k) as its input and has to decide whether (I, k) ∈ L for some parameterized problem L. Many parameterized algorithms work in two stages: reduction to a problem kernel and bounded search tree. Their time complexity is then of the form O(p(|I|) + q(k)ξ), where q(k) is the size of the problem kernel. We show how to modify these algorithms to obtain time complexity O(p(|I|)+ξ), if q(k) is polynomial.