The complexity ecology of parameters: An illustration using bounded max leaf number

Abstract

In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-COLORING or HAMILTON PATH or MINIMUM DOMINATING SET for graphs of bounded max leaf number? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input of bounded max leaf number, based on the polynomial-time extremal structure theory associated to the parameter max leaf number. (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach. © Springer-Verlag Berlin Heidelberg 2007.