We describe some new, simple and apparently general methods for
designing FPT algorithms, and illustrate how these can be used to
obtain a significantly improved FPT algorithm for the Maximum Leaf
Spanning Tree problem. We sketch how the methods can be applied to
a number of other well-known problems, including the parametric dual
of Dominating Set (also known as Nonblocker), Matrix Domination,
Dominating Set for Line Graphs and Feedback Vertex Set for
Undirected Graphs. The main payoffs of these new methods are
in improved functions f(k) in the FPT running times, and in general
systematic approaches that seem to apply to a wide variety of
problems.