TY - JOUR
T1 - The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number
AU - Fellows, Michael R.
AU - Lokshtanov, Daniel
AU - Misra, Neeldhara
AU - Mnich, Matthias
AU - Rosamond, Frances
AU - Saurabh, Saket
PY - 2009
Y1 - 2009
N2 - 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 ml(g) 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, hamilton path, minimum dominating set, minimum bandwidth or many other problems, for graphs of bounded max leaf number? what optimization problems are w[1]-hard under this parameterization? we do two things: (1) we describe much improved fpt algorithms for a large number of graph problems, for input graphs g for which ml(g)=k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (fpt) runtimes o *(f(k)). (2) the way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.
AB - 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 ml(g) 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, hamilton path, minimum dominating set, minimum bandwidth or many other problems, for graphs of bounded max leaf number? what optimization problems are w[1]-hard under this parameterization? we do two things: (1) we describe much improved fpt algorithms for a large number of graph problems, for input graphs g for which ml(g)=k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (fpt) runtimes o *(f(k)). (2) the way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.
U2 - 10.1007/s00224-009-9167-9
DO - 10.1007/s00224-009-9167-9
M3 - Article
SN - 1432-4350
VL - 45
SP - 822
EP - 848
JO - Theory of Computing Systems
JF - Theory of Computing Systems
IS - 4
ER -