Coordinatized Kernels and Catalytic Reductions: An Improved FPT Algorithm for Max Leaf Spanning Tree and Other Problems Michael R. Fellows 1 , Catherine McCartin 1 , Frances A. Rosamond 1 , and Ulrike Stege 2 1 School of Mathematical and Computing Sciences, Victoria University Wellington, New Zealand {Mike.Fellows,Catherine.Mccartin,Fran.Rosamond}@mcs.vuw.ac.nz 2 Department of Computer Science, University of Victoria Victoria, B.C., Canada stege@csr.uvic.ca Abstract. We describe some new, simple and apparently general meth- ods 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. Furthermore, we sketch how the methods can be applied to a number of other well-known problems, including the para- metric dual of Dominating Set (also known as Nonblocker), Ma- trix Domination, Edge Dominating Set, 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. 1 Introduction The investigations on which we report here are carried out in the framework of parameterized complexity, so we will begin by making a few general remarks about this context of our research. The subject is concretely motivated by an abundance of natural examples of two different kinds of complexity behaviour. These include the well-known problems Min Cut Linear Arrangement, Bandwidth, Vertex Cover, and Minimum Dominating Set (for defini- tions the reader may refer to [GJ79]). All four of these problems are NP-complete, an outcome that is now so rou- tine that we are almost never surprised. In the classical complexity framework that pits polynomial-time solvability against the ubiquitous phenomena of NP- hardness, they are therefore indistinguishable. All four of these decision problems take as input a pair consisting of a graph G and a positive integer k. The positive integer k is the natural parameter for all four problems, although one might also wish to consider eventually other problem parameterizations, such as treewidth. We have the following contrasting facts:  Ulrike Stege is supported by the Pacific Institute for the Mathematical Sciences (PIMS), where she is a postdoctoral fellow. S. Kapoor and S. Prasad (Eds.): FST TCS 2000, LNCS 1974, pp. 240–251, 2000. c  Springer-Verlag Berlin Heidelberg 2000